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STATISTICAL THEORY FOR HYDROACOUSTIC AND MARK-RECAPTURE ESTIMATION OF FISH
POPULATIONS
B.M. Sarath Gamini Banneheka
B.Sc.(Hons.), University of Sri Jayewardenepura, Sri Lanka, 1985
M.Sc.(Mathematics), King's College London, University of London, England, 1987
M.Sc.(Statistics), Simon Fraser University, Canada, 1991
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF T H E REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in the Department
of
Mathematics and Statistics
@ B.M. Sarath Gamini Banneheka 1995
SIMON FRASER UNIVERSITY
May 1995
All rights reserved. This work may not be
reproduced in whole or in part, by photocopy
or other means, without the permission of the author.
APPROVAL
Name:
Degree:
Title of thesis:
B.M. Sarath Gamini Banneheka
DOCTOR OF PHILOSOPHY
STATISTICAL THEORY FOR HYDROACOUSTIC AND
MARK-RECAPTURE ESTIMATION O F FISH POPULA-
TIONS
Examining Committee: Dr. G.A.C. Graham
Chair
Dr. Richard D. Routledge
Senior Supervisor
Dr. Charmaine B. Dean
Dr. Richard A. Lockhart
Dr. Carl J. Schwarz
l ~ $ John R. Skalski, University of Washington
External Examiner
Date Approved: May 19, 1995
PARTIAL COPYRIGHT LICENSE
I hereby grant to Simon Fraser Universi the right to lend my
2 P thesis, pro'ect or extended essay (the title o which is shown below) to users o the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. I further agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. I t is understood that copying or publication of this work for financial gain shall not be allowed without my written permission.
Title of T h e s i s / P 5
S t a t i s t i c a l Theory f o r Hydroacoustic and Mark-Recapture
Est imation of F ish Populat ions
Author:- (signature)
B.M. Sa ra th Gamini Sanneheka
May 19 , 1995.
(date)
Abstract
Hydroacoustic sampling is a widely used technique for the estimation of fish population
sizes. The need to estimate the volume of the hydroacoustic beam makes this method
difficult. Recent research has focused on overcoming this difficulty. Existing techniques,
called 'duration-in-beam methods' estimate the volume of the beam assuming it is cone
shaped.
We present a hydroacoustic sampling technique that captures the advantages of the
duration-in-beam method, but avoids the cone-shaped assumption. The method was specif-
ically developed for making in-river estimates of the abundance of migrating fish, and has
also been adapted to estimate resident fish populations. The population sizes in these cases
can be written as sums of products of pairwise means. The problem of constructing approx-
imate confidence intervals for such quantities is examined. Specific theory for normal data
are provided and suggestions are made for other situations.
Tag recovery methods are also commonly used to estimate fish (and other animal) popu-
lations. Two sample tag recovery experiments when the tagging and recovering are stratified
have been studied by several authors, who have provided maximum likelihood estimates and
moment type estimators. We enrich this class of estimators by introducing the least squares
estimates which are applicable when the numbers of tagging and recovery strata are unequal.
The asymptotic variances of these estimates are also provided.
A common practice in stratified tag recovery experiments is to pool the strata before or
after the experiment. This can produce inconsistent estimates. Sufficient conditions for the
consistency of the estimates in the case of complete pooling are already known. We examine
sufficient conditions for the case of partial pooling.
iii
Acknowledgments
I would like to express my sincere appreciation and gratitude to my senior supervisor Prof.
Richard Routledge for providing constant encouragement and invaluable guidance with much
patience throughout the course of my study. I feel very fortunate to have him as my
supervisor. My thanks also go to Prof. Carl Schwarz who suggested the problems in
Chapters 5 and 6, and provided suggestions for their solution.
I will always remember the Department of Mathematics and Statistics of Simon Fraser
University as a friendly place where I received help whenever needed. Specifically, I consider
the opportunity to have known and worked with Professors Richard Routledge, Charmaine
Dean, David Eaves, Richard Lockhart, Michael Stephens, Carl Schwarz, Tim Swartz, and
Larry Weldon as an asset. I immensely value the education that I received from them
and gratefully acknowledge the considerable help provided to me. I also wish to thank
Sylvia Holmes, Maggie Fankboner, Dianne Pogue, and other department staff. The financial
support provided by the Department of Mathematics and Statistics, and from research
grants received by Prof. Routledge is greatly appreciated.
I would like to extend my thanks to Ian Guthrie and Jim Woodey of the Pacific Salmon
Commission, for helping me understand the practical side of the problem discussed in Chap-
ter 3 and for providing necessary information to solve the problem.
Finally I thank all my colleagues whose help and friendship made my stay at Simon
Fraser University comfortable. A special thanks to my dear friend and colleague Chandanie
Perera, without whose kind, caring support, the study and stay in Canada would have been
much difficult.
Dedication
To my parents
Contents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract iii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments iv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication v
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables ix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures xi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1
. . . . . . . . . . . . . . 2 Use of Echo Sounding Techniques for Detecting Fish 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Terminology 5
2.1.1 Echo Sounder . . . . . . . . . . . . . . . . . . . . . . . . . . 5
. . . . . . . . . . . . . . . 2.1.2 Echo Trace (Target), and Echogram 7
2.1.3 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
. . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Directivity Pattern 7
2.1.5 Targetstrength . . . . . . . . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Effective Beam 9
. . . . . . . . 2.2 Estimation of Beam Volume: Duration-in-Beam Method 10
3 Estimation of Fish Populations Using Echo Sounding Techniques . . . . . . . 13
. . . . . . . . . . . . . . . 3.1 Estimation of Migrating Fish Populations 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Methods 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Assumptions 19
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Estimator 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Bias 29
. . . . . . . . . . . . . . . . . . . . . 3.1.5 Estimation of Variance 32
. . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Numerical Example 33
3.2 Estimation of Resident Fish Populations . . . . . . . . . . . . . . . . . 35
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Estimator 35
. . . . . . . . . . . . . . . . . . . . . 3.2.2 Estimation of Variance 39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Discussion 40
4 Confidence Intervals for a Sum of Pairwise Products of Means . . . . . . . . 42
. . . . . . . . . . . . . . . . . . . . . . . . 4.1 Notation and Assumptions 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Notation 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Assumptions 45
. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Normally Distributed Data 45
. . . . . . . . . . . . . . . . 4.2.1 Likelihood Ratio Approach '. . . 45
. . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Wald Test Approach 60
. . . . . . . . . . . . . 4.3 Adaptation to Distributions other than Normal 64
. . . . . . . . . . . . . . . 4.4 Worked Example and Monte Carlo Studies 67
. . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Worked Example 67
. . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Monte Carlo Studies 69
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary 75
. . . . . . 5 Stratified Two-Sample Tag Recovery Census of Closed Populations 77
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Notation 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Estimation 82
. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Assumptions 86
5.2.2 Estimation of the Numbers of Animals in the Recovery Strata
(V) 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Estimation of the Numbers of Animals in the Tagging Strata
(U) 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Some Remarks 91
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Consistency 96
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Variance 97 A
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Variance of 97
vii
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Variance of 6 103
. . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Relaxing Assumption 4 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Discussion 110
. . . . . . . . . . . 6 Pooling in a Stratified Two-Sample Tag Recovery Census 112
. . . . . . . . . . . . . . . . . . . . . 6.1 Partial Pooling of Tagging Strata 114
. . . . . . . . . . . . . . . . . . . . 6.2 Partial Pooling of Recovery Strata 118
. . . . . . . . . . . . . 6.3 Partial Pooling of Tagging and Recovery Strata 122
. . . . . . . . . . . . . . . . . . . . . . 6.4 Variances of Pooled Estimates 123
. . . . . . . . . . . . . . . . . . . . . . . . 6.5 Biases of Pooled Estimates 128
. . . . . . . . . . . . . . . 6.6 Worked Examples and Monte Carlo Studies 131
. . . . . . . . . . 6.6.1 Example 1: Schaefer's sockeye salmon data 131
. . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Monte Carlo Studies 136
. . . . . . . . . . . . 6.6.3 Example 2: Schwarz's pink salmon data 139
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Discussion 144
. . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Some Useful Results 145
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 147
viii
List of Tables
Summary statistics of the data collected by the Pacific Salmon Commission
on August 31,1993, a t Mission, B.C. Also shown are the estimated numbers of
fish migrating upriver per second and the corresponding estimated variances. 34
Summary statistics of the data collected by the Pacific Salmon Commission
on August 31, 1993, a t Mission, B.C. . . . . . . . . . . . . . . . . . . . . . . 65
95% confidence intervals for the fish problem . . . . . . . . . . . . . . . . . . 69
Group numbers for combinations of distributions . . . . . . . . . . . . . . . . 70
Group numbers for combinations of sample sizes . . . . . . . . . . . . . . . . 70
Observed coverage in 10,000 simulations: E(x1)=(1.20, 6.38,1.74), V(x1)=(1 .53 ,
17.4, 2.36), E(x2)=(1.07, 0.76, 0.79), V(xz)=(0.136, 0.066, 0.130). . . . . . . 71
Observed coverage in 10,000 simulations: E(x1)=(1.20, 6.38,1.74), V(xl)=(1.53,1
7.4, 2.36), E(x2)=(1.07, 0.76, 0.79), V ( x 2 ) = ( 0.136, 0.066, 0.130). . . . . . . . 72
Observed coverage in 10,000 simulations for normal data. (Nominal coverage
=95%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Observed coverage in 10,000 simulations for normal data. (Nominal coverage
=95%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Observed coverage in 5,000 simulations for normal data. (Nominal coverage
=95%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Symbols for Numbers of Animals . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 Schaefer's data on sockeye salmon: n.j. m; and vj for s = 8. t = 9 (from
Darroch [1961:Table 11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Schaefer's data on sockeye salmon: n,rj. mf and vJ for s* = 4. t* = 4 (from
Darroch [1961:Table 21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 Darroch's Estimates of V* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Our Estimates of V* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Our Estimates of U* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.6 Fraser River pink salmon data: Areas 1-3 . . . . . . . . . . . . . . . . . . . . 140
6.7 Fraser River pink salmon data: Areas 4-6 . . . . . . . . . . . . . . . . . . . . 141
6.8 Fraser River pink salmon data: Recovered numbers of marked fish ( after
pooling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . 6.9 Estimated numbers of unmarked fish a t the tagging stage 143
. . . . . . . . . . . 6.10 Estimated numbers of unmarked fish at the tagging stage 143
List of Figures
2.1 (a) Echosounder (b) Sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Functioning of an echo-sounder . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Three-dimensional view of the directivity . . . . . . . . . . . . . . . . . . . . 8
3.1 Schematic diagram showing hydroacoustic sampling activities on the Fraser
River a t Mission (courtesy of the Pacific Salmon Commission, Vancouver). . . 16
3.2 An echogram from a transect sounding (courtesy of the Pacific Salmon Com-
mission, Vancouver) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 An echogram from a stationary sounding (courtesy of the Pacific Salmon
Commission, Vancouver). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The distance that a fish swims to cross through the acoustic beam (C) can
be calculated from the radius r(z) of the beam at depth z and from the half
angle 8 formed in the center of the beam . . . . . . . . . . . . . . . . . . .
4.1 Behavior of p(X) and S(X) when 5152 > 0 (example calculation using nl =
3 0 , n z = 3 0 , f l = 1 0 , ~ 2 = 2 0 , s l = 3 a n d s 2 = 5 ) . . . . . . . . . . . . . . . . 4.2 An example where S(X;) < X:,(l-,): (example calculation using nl = 30,722 =
30, Z1 = 10, Z2 = 20, s1 = 15, s 2 = 5 and a = .05. Vertical dotted line repre-
sents X = 12. Horizontal dotted line represents x:,.~~.) . . . . . . . . . . . . 4.3 Stem and leaf plots of target widths, W;'s (based on hydroacoustic data
collected by the Pacific Salmon Commission on August 31, 1993 at Mission,
B.C.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Histograms of Ni's and Mi's (based on hydroacoustic data collected by the
Pacific Salmon Commission on August 31. 1993 at Mission. B.C.). The height
of each bar is the proportion of observations in the corresponding interval on
the horizontal axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
. . . . . . . . 4.5 Calculation of 95% confidence interval for p = c : ~ ~ E(N,)E(M.) 68
6.1 Percent relative error of & in three Monte Carlo studies (each study is based
on 1000 simulations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
xii
Chapter 1
Introduction
Estimation of the sizes of animal populations is required for many purposes such as wildlife
management, fisheries and pest control. Recently, there has been a growing interest in this
branch of statistics. Advances in general statistical theory and instrumentation have created
opportunities for substantive improvements.
Hydroacoustic sampling and tag-recovery studies are two widely used methods in esti-
mating fish populations. For example, these methods are used for the estimation of sockeye
(Oncorhynchus nerka) and pink (Oncorhynchus gorbuscha ) salmon in the Fraser River,
Canada. In hydroacoustic sampling, a beam of sound waves is transmitted through water.
The number of fish detected by the beam is then extrapolated to estimate the fish popu-
lation. In-river hydroacoustic sampling provides quick daily estimates of abundance in the
Fraser. Quick estimation is essential for efficient management of fisheries. Tag-recovery ex-
periments in spawning areas provide estimates of numbers of fish reaching spawning grounds.
These estimates are useful in post-season evaluation of the management programme, and in
monitoring the dynamics of the stocks.
With the development of new data collection techniques such as hydroacoustic sampling
methods, radio telemetry, remote sensing etc., the need arises for suitable statistical methods
Chapter 1. Introduction 2
to extract the information from such data. The first half of this thesis focuses on hydroa-
coustics. Chapter 2 provides a brief introduction to hydroacoustic sampling techniques.
Our interest in this area was initiated by a request made by the Pacific Salmon Commis-
sion. They needed to gauge the variance of a hydroacoustic abundance estimator developed
by the International Pacific Salmon Fisheries Commission (IPSFC) in 1977. This method
had been used by the IPSFC (from 1977 until 1986) and by its successor, the Pacific Salmon
Commission (since 1986) for estimating the abundance of Fraser River sockeye and pink
salmon that pass Mission, British Columbia, during their upstream migration to spawning
grounds.
We refined the formula for the estimator itself, with the aim of improving its statistical
behavior while maintaining the practical advantages of the existing method. In Chapter 3,
we describe the derivation of this estimator. As well, formulae are derived for the bias and
the variance of the estimator. This method is now being tested for full-scale implementation
by the Pacific Salmon Commission. In Chapter 3, we also suggest suitable modifications of
the method to estimate resident fish populations.
The quantities we estimated can be written in terms of sums of products of pairwise
means. We consider the problem of constructing confidence intervals for such quantities
in Chapter 4. The theoretical results are derived formally for constructing approximate
confidence intervals for normally distributed data. However, often the sample sizes in these
kind of experiments are large, and a central limit theorem can be used to adapt the normal-
based results for non-normal distributions. The validity of the theoretical results derived in
this chapter is examined by a Monte Carlo study. We end Chapter 4 with a description of
the Monte Carlo study.
Tag recovery experiments are widely used in estimating animal populations. In the sim-
plest kind of tag-recovery experiments, a random sample of animals is tagged and released.
After allowing for the tagged animals to disperse, another random sample is taken from the
population and the number of tagged animals in the sample is recorded. These experiments
Chapter 1. Introduction 3
are therefore often called two-sample tag-recovery experiments. When the population is
stratified geographically or temporally, this feature can be exploited to obtain more precise
estimates of the population size. In this method, a known number of tagged animals is re-
leased in each stratum, using a different tag for each stratum. After allowing for the tagged
animals to disperse, a random sample is taken from each recovery stratum. The numbers of
untagged animals and the numbers of tagged animals of each type in each sample are then
recorded. These experiments are often called stratified tag-recovery experiments. Maximum
likelihood estimators and moment-type estimators have already been developed for estimat-
ing population sizes using stratified tag-recovery data. When the number of tagging strata
and recovery strata are unequal, these require imposing restrictions on unknown survival
probabilities to obtain the estimates. In Chapter 5, we derive least squares estimates that
avoid this requirement. We also prove the consistency of the derived estimates and provide
formulae for the variances of the estimates.
In certain situations, the experimenter is unable to use different marks for different
strata, is unaware of how the population is stratified, or, after the collection of the recovery
data, finds only a very small number of a certain type of tags. A common practice in such
situations is to pool the strata before or after the experiment. This can produce inconsistent
estimates. Sufficient conditions for the consistency of the estimates in the case of complete
pooling are already known. In Chapter 6, we discuss the estimation problem and provide
conditions for consistent estimates to be produced after partial pooling.
Chapter 2
Use of Echo Sounding Techniques
for Detecting Fish
In hydroacoustic sampling, a beam of sound waves is transmitted through water. The
number of fish detected by the beam is used to estimate the total number of fish in a
given volume, or the total number of fish that pass a given point. A hydroacoustic sampling
method for the detection of fish was first reported in the scientific literature in 1929 (Kimura
[19]). In this experiment, continuous waves at a frequency of 200kHz were directed across
ponds containing goldfish. Crittenden [ll] provided a review of the historical background
up to 1970.
The purpose of this chapter is to introduce the terminology used in the next chapter, and
to provide a brief summary of the 'duration in beam method,' that is used to estimate the
volume of the hydroacoustic beam. Further details can be found in Burczynski [6], Thorne
[39], [40], Johannesson and Mitson [16], and Crittenden [ll].
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish
2.1 Terminology
2.1.1 Echo Sounder
Generally speaking, an apparatus used for obtaining information about underwater objects
and cvcnts with thc usc of sound wavcs is callcd a 'sonar system". Sonar systcms uscd in
fisheries work produce ultrasounds, i.e. sounds with a frequency usually ranging from 12 to
500 kHz (which are not detectable by a human ear). According to Burczynski [6], a sonar
system that transmits vertically is usually called an 'echo-sounder' (Figure 2.1 a) while a
sonar system that transmits horizontally is called 'a sonar' (Figure 2.1 b).
Figure 2.1: (a) Echosounder (b) Sonar
'sonar is an abbreviation for Sound Navigation and Ranging
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish 6
An echo-sounder consists of four main components (Figure 2.2), the transmitter, the
transducer, the receiver, and the recorder. The function of the transmitter is to produce
energy in the form of pulses of electrical oscillations. A pulse is generated when a timer
activates the electrical transmitter a t a known frequency for a fixed period of time. Usually
this frequency is 38kHz to 120 kHz. The time interval during which the transducer actually
vibrates in generating each pulse, is called the 'pulse duration'. Typically this is about one
millisecond. The electrical oscillations thus generated are then converted mechanically into
sound waves in the water a t the vibrating face of the transducer, which continues to generate
sound until the timer switches off the transmitter. The result is a sound pulse of the same
frequency, traveling through the water away from the face of the transducer. The number
of pulses (or transmissions) sent out per unit time is called the 'pulse repetition rate'.
time '1$4- Display w
Receiver i-i
- Target
Figure 2.2: Functioning of an echo-sounder
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish 7
After each pulse the system waits for a certain period to receive echoes from targets
(e.g. fish) in the ensonified volume of water. For example, when a transducer is operating
at a pulse repetition rate of sixty transmissions per minute, it generates sound for one
millisecond in the transmitting mode, then waits for 999 milliseconds in the receiving mode,
then generates the next one-millisecond pulse and so on. The transducer, when in the waiting
mode, performs the reverse of its function in the transmitting mode, i.e. it converts pressure
oscillations produced at its face by the echo, into electrical oscillations. The function of the
receiver is to amplify these oscillations so that they can be recorded or displayed.
2.1.2 Echo Trace (Target), and Echogram.
The detectable sign on the display.unit is called the 'echo trace'. The record of echo traces is
called the 'echogram' (see Figures 3.2 and 3.3). A trained technician can usually distinguish
echo traces of fish, from those of other objects. In the next chapter, we loosely use the term
'target' for an echo trace of a fish.
2.1.3 Range
The distance between the fish and transducer can be calculated by multiplying half the time
interval between transmitting the pulse and receiving its echo by the sound velocity. The
average distance of the fish from the transducer at the first detection and the last detection
is called the 'range' of the fish.
2.1.4 Directivity Pattern
A transducer can be regarded as an array of point sources of sound. A point source radiates
sound uniformly in all directions. However if two point sources that generate the same
sound are located fairly close to one another, the resulting sound intensity will not be the
same in all directions. In some directions, the waves from the two sources will be in phase or
Chapter 2. Usc of Echo Sounding Tcchniqucs for Dctccting Fish 8
ncarly so, and will rcinforcc cacll othcr to producc a high dcnsity; in othcr dircctions, tlicy
will canccl cach otllcr. This phcnomcnon, callcd intcrfcrcncc, produccs a regular but non-
uniform distribution of sound intcnsity according Lo dircction. T l ~ c intcnsity distribulion of a
trnnsduccr has a masimum on its axis ( a linc passing tlwougl~ its ccntrc pcrpcndicular to thc
surfacc) and lowcr vducs clscwl~crc. I t is convcnicnl to tlcscribc lllc intcnsity distribution,
or 'tlircclivily ~ m l l c r ~ l ' , of ;L Lr;w~stl~~ccr ;IS lhc rcl;~livc i~ilcnsily ill cad1 bearing ( $ , 0 )
(scc Figurc 2.3) will1 rcspccl to i~~lcnsi ty Io,o, i ~ l o ~ i g 111c ;&xis. Thc ri~lio 01 lhc lwo inlcnsilics
a t a givcn range is conslant. This ratio is dcnprcd by 6(), 0 ) = e.
b Y \ Transducer
Figure 2.3: Thrcc-tli~rlcnsiolld vicw 01 tllc dircctivity
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish
According to Clay and Medwin [8], a cylindrical transducer produces a directivity pattern
that is symmetrical about the acoustic axis. Figure 2.3 shows a typical directivity pattern
of a circular transducer. Intensity is highest along the axis. With increasing deviation from
the axis, intensity first decreases to a minimum, then increases again to a value much lower
than the maximum on the axis, then decreases and so on. The region of high values near
the axis is called the main lobe of the beam and the successively smaller peaks around the
main lobe are called side lobes.
2.1.5 Target Strength
Experiments have shown that fish species with swim bladders reflect about 85 percent of the
sound energy by the swim bladder. The ratio of echo intensity from a target to the incident
intensity is called the 'target strength'. The target strength (or scattering cross section) of a
fish has a directivity pattern quite similar to that of a transducer. In general the directivity
pattern depends on the anatomy of the fish, its overall size, and the dimensions of the swim
bladder relative to the wave length. For a given species of fish, and for a given wavelength,
there is a close relationship between the target strength and the size of the fish; the larger
the fish, the larger its target strength. The relative intensity of sound reflected by a fish
back to the source also depends on its orientation relative to the source.
2.1.6 Effective Beam
As a sound signal propagates through water, its intensity decreases with distance. There
are two reasons for this: the dispersal of energy due to spreading effect, which is called the
geometric loss, and the temperature and frequency-dependent power loss, which is called
the absorption loss. The sum of these two types of energy loss is called the transmission
loss. The intensity of the signal is further reduced according to reflective properties of the
target and by the subsequent transmission loss of the echo. According to Thorne [40], the
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish
intensity of the signal (I,) when it returns to the transducer can be described as
where
target strength,
intensity of the signal a t a unit range from the transducer,
absorption coefficient (a function of signal frequency, water temperature
and salinity),
distance to the target, and
directivity pattern.
The absorption coefficient a is negligible for short or moderate ranges in fresh water.
Therefore, if the receiving intensity threshold of the transducer is T, the maximum detection
range for a fish of given target strength k, a t bearing (+,9) can be approximated by
The volume defined by this (R, +,9) relationship given by (2.2) is the 'effective beam' for
fish of target strength k. It is usually hoped that the targets encountered in side lobes will
return echoes that are below the detection threshold. Then, for a cylindrical transducer the
effective beam can be approximated by a single lobe that is radially symmetrical about the
acoustic axis.
2.2 Estimation of Beam Volume: Duration-in-Beam Met hod
Most hydroacoustic surveys attempt to estimate the number of fish in a population by
extrapolating the number of fish counted in the acoustic beam to the entire survey region
according to the ratio between the volume of water in the survey region and the volume
sampled by the acoustic beam (e.g., Thorne and Dawson [38]). A similar method is used by
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish 11
Skalski et al. [36] to estimate fish passage at a dam using a weighted sum of the numbers
of fish ensonified by a beam at different depth ranges. The weights are determined by the
dimensions of the beam and the dam passage ways. A major difficulty with these methods .
is that the volume of the acoustic beam must be estimated. Not only is the shape of the
beam complicated by interference patterns that can produce side lobes (Clay and Medwin
[S], pp. 144- 146; MacLennan and Simmonds [24], pp. 13-20; and references therein), but also
the effective dimensions of an acoustic beam vary with the size, orientation and swimming
movements of fish that encounter the beam (Clay and Medwin [S], 245; MacLennan and
Simmonds [24], 137; and references therein). Dependence on estimates of beam width also
creates problems with enumerating populations that have a wide distribution of body sizes.
Usually the effective beam is assumed to be a simple cone with an unknown angle.
One popular method for estimating the effective sampling beam volume is the empirical
technique called the 'duration-in-beam method'. This method was first conceived by R.
Thorne and H. Lahore in 1970 and first applied to estimation of juvenile sockeye salmon
in Lake Washington in 1971. (Thorne [39], Thorne and Dawson [38]). Since then several
authors have derived alternative estimates based on the same principle (e.g. Nunnallee and
Mathisen [27], Crittenden et al. [12], and Kieser and Ehrenberg [IS]).
According to Thorne [39], the effective diameter of the beam at a given depth is estimated
from the number of successive echoes received from individual fish ('duration-in-beam') as
the transducer moves over the fish at a known speed. Any movement of fish is assumed
to be negligible compared to the boat speed. His estimate of the diameter at range R is
calculated as
where e(R) is the average number of echoes from individual fish at range R, v is the pulse
repetition rate and u is the boat speed. The beam angle and the volume can now be
estimated by simple trigonometry from the beam diameter and the range.
Nunnallee and Mathisen [27] proposed an unbiased estimator for the sine of the half angle
Chapter 2. Use of Echo Sounding Techniques for Detecting Fish 12
of the assumed cone-shaped beam. This estimator is based on the relationship between the
expected value of the
transducer to the fish.
where $ is the sine of
number of echoes from individual fish and the range R from the
They found that
the half beam angle. They considered several range strata, and for
each stratum, $ was estimated by ~ E U / T V R . Then, taking the average of these estimates
over strata (say a strata), $ was estimated by
Crittenden et al. [12] suggested an alternative estimator based on weighted regression.
They noticed that equation (2.3) can be expressed in the form of a model that is linear in
R, has slope /3 = e, and passes through the origin,
Finding that the variance of e is proportional to R2, they regressed e on R with weights
w = 1/R2. Then, based on the estimated slope of this regression, they estimated $ by
- 221 C w;e;R; 2u $ = - = - Average(e/ R).
nu E w;Rq nu
These methods represent a substantial reduction in the information needed to estimate fish
abundance, because the calibration and enumeration are essentially done simultaneously
using real targets. Nunnallee and Mathisen's [27] estimate for sine of the cone angle leads
to an unbiased estimate of beam volume. Crittenden et al.'s [12] estimate for sine of the
cone angle leads to a minimum variance unbiased estimate of the beam volume. However,
these estimates do not lead to a direct unbiased estimate of fish population size because the
population size is not a linear function of this quantity. It is a function of the the reciprocal
beam volume. In the next chapter, we present an alternative method that requires only that
the beam be circular symmetric and does not require to estimate the volume by duration-
in-beam method.
Chapter 3
Estimation of Fish Populations
Using Echo Sounding Techniques
3.1 Estimation of Migrating Fish Populations
The purpose of this section1 is to present a hydroacoustic sampling technique that captures
the advantages of the duration-in-beam method, but that is particularly suitable for making
in-river estimates of the abundance of migrating fish. The technique was developed by the
International Pacific Salmon Fisheries Commission (IPSFC) in 1977, and used by the IPSFC
from then until 1986, and by its successor, the Pacific Salmon Commission, since then. The
application was the estimation of abundance of Fraser River sockeye (Oncorhynchus nerka)
and pink (Oncorhynchus gorbuscha ) salmon that pass Mission, British Columbia, during
their upstream migration to spawning grounds in the Fraser River watershed (Woodey
[42]). We have refined the calculation methods to improve the statistical properties of the
estimator and have developed formulae for estimating the variance and bias of the estimate.
With this method, a boat-mounted echo sounder that conducts shore-to-shore transects
'A paper based on the contents of this section has been submitted to the Canadian Journal of Fisheries and Acquatic Sciences as: Banneheka, S.G., R.D. Routledge, I.C. Guthrie, and J . C. Woodey. 1995. Estimation of In-River Fish Passage Using a Combination of Transect and Stationary Hydroacoustic Sampling.
13
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
across the river is used to estimate the density of fish targets per depth interval. Stationary
soundings provide an estimate of the amount of time it takes fish in each depth interval to
swim through the beam. Combining these estimates yields estimates of the number of fish
that pass the transect location per unit time. The method eliminates the need to estimate
beam dimensions directly, as long as the same hydroacoustic equipment and settings are
used for both the transect and stationary soundings. The sampling techniques used by the
Pacific Salmon Commission are further described in the following section.
3.1.1 Methods
The hydroacoustic program conducted by the Pacific Salmon Commission in 1993 at Mission,
British Columbia, is briefly described below. Minor technical aspects of the methods varied
over time as, for example, the abating spring freshet led to declining river depths. The
specific methods described below were applied on August 31. Data from this day are later
used in a numerical example illustrating the calculations.
Equipment
The equipment that was used for both the transect and stationary soundings was in-
stalled on a 5.8 m aluminum vessel. A BioSonics Model 105 echo-sounder operating at a
frequency of 50 kHz was used with a 34 degree (full angle) circular-beam transducer. The
transducer was mounted in an aluminum stabilizer and towed from a davit 1.1 m from the
starboard gunnel at a depth of 0.5 m. Settings for the equipment included a pulse rate of
15.3 ensonifications per second and, to compensate for two-way spreading loss, a time-varied
gain of 40 Log R (where R = range of target). To record the soundings, a BioSonics Model
111 Thermal Chart Recorder was used. It was set to record a 15 m depth range with 5 m
depth intervals marked. This depth range encompassed the maximum river depth at the
site, except during spring freshet, when the depth range was increased to 20 m. Paper speed
for the recorder was 0.478 mm s-'. The equipment was serviced annually, and each day the
crew checked the settings to ensure that they had not been accidentally changed.
Chapter 3. Estimation o f Fish Populations Using Echo Sounding Techniques 15
Site
The data collection site was about 1 km upstream of the railway bridge across the Fraser
River at Mission, British Columbia. River width at this site is about 437 m. This location
has been used by the IPSFC and the Pacific Salmon Commission for echo sounding since
1977 because it satisfies several criteria. First, fish actively migrate at this location: the
site is in a straight stretch of the river with no eddies or deep pools where fish can hold or
mill about. Second, it has a rapid drop-off to deep water on both shores: there are only a
few meters along each shore that cannot be sampled, and few sockeye salmon are believed
to migrate this close to shore. Third, the smooth bottom is unlikely to acoustically obscure
fish. Fourth, the flow of the river here is relatively even and moderately slow so that a
small boat can cross the river at suitable speeds for echo sounding without being swept
downstream. Finally, the site is close to a road, dock and fuel source.
Field Sampling
Two data collection activities were performed (Figure 3.1). The first consisted of 215
shore-to-shore transect soundings across the river, perpendicular to the direction of fish
movement. All transects used the same transect line, boat speed, and echo-sounder settings.
These transect soundings took about five minutes each to complete.
Second, nine stationary soundings were conducted along the transect line, using the
same equipment and settings used for the transect soundings. These soundings were taken
at scheduled times, with one located in each of the south, center and north sections of the
transect line during each 8-hour shift. The boat was anchored during these soundings.
Hydroacoustic data were collected over a twenty-four hour cycle, from 5:00 a.m. to 4:59
a.m. the next morning. Transect soundings were conducted for about eighteen hours, and
stationary soundings for about three hours. Other activities such as refuelling, changing
crews and setting anchors for the stationary soundings took up the remaining three hours.
Chapter 3. Estimation oiFis'ish Populations Using Echo Sounding Tecl~niques
South Shore North Shore
1 I I I
0 100 200 300 1100 ' .
Disbncc Across Rivcr (m) . . . . . .
Figure 3.1: Schematic diagram showing hydroacoustic sampling activities on the kaser
River at Mission (courtesy of the Pacific Salmon Commission, Vancouver).
Data processing
After the 24-hour recording session, the resulting echogram was interpreted by a trained
hydroacoustics biologist. For each transect, the number of fish targets in each depth stratum
were counted (see Figure 3.2). For each stationary sounding, the widths of each echo traccs
from fish (target widths) in each depth stratum were mcasurcd using a micrometer (see
Figurc 3.3). Periodically, thc papcr spccd of thc rccordcr was vcrified, using marks that thc
echo-sounder recorded on the echogram cach minute.
Chapter 3. Estimation of fish Populations Using Echo Sounding Techniques 17
Targets (echo traces of fish)
Figure 3.2: An echogram irom a transect sounding (courtesy of the Pacific Salmon Com-
mission, Vancouver)
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
Targct Width :
I)
Targets (echo traces of fish) . . I
Figure 3.3: An echogram from a stationary sounding (courtesy of the Pacific Salmon Com-
mission, Vancouver).
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 19
3.1.2 Assumptions
1. If there are fish present that are not part of the population of interest (e.g., other
species), then these can either be distinguished on the record or otherwise removed
from the estimates. (In the Fraser River programme, the number of resident [i.e.,
non-migratory] fish was estimated by sampling when few salmon were in the river
[i.e., before or after salmon runs or after commercial gillnet fisheries downstream].
The target density associated with resident fish was then removed from the daily
total target density to obtain an estimate of daily salmon density, which was used to
estimate the salmon population. To determine the sockeye salmon component in this
estimate, the species composition of these co-migrating salmon was obtained from
a test fishery using a variable-mesh gillnet that was conducted downstream of the
Mission site.)
2. The beam has circular horizontal cross sections. (This assumption is less stringent than
the more common requirement that the beam be a cone, and is required because the
presence of small side lobes near the transducer caused by local interference patterns
(Clay and Medwin [8], 144-146; MacLennan and Simmonds [24], 13-20; and references
therein) cannot be ignored, particularly in the shallow water near the shorelines at
the sampling site.)
3. The hydroacoustic beam extends to the river bottom. (In the Mission echo-sounding
programme, the depth range recorded by the echo-sounder encompassed the maximum
depth encountered at the study site.)
4. The records of fish echos do not overlap on the record. (Schooling or heavy runs could
generate high local abundances that would lead to substantial numbers of overlapping
records. In extreme cases, fish density can be high enough to extinguish the bottom
echo signal [Foote 19901. In the present application, the local abundance of sockeye
salmon was not large enough to cause significant overlapping of fish echoes.)
5. Fish echos can be distinguished from the background noise. (The ability to distinguish
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 20
fish from noise is determined by the experience of the interpreter and the selection of
criteria that can be consistently applied to determine whether a target is "real" [e.g.,
three successive echoes].)
6. Fish are distributed randomly with a uniform probability density over depth within
each stratum. (The validity of this assumption, along with the following one, can be
promoted by choosing suitably narrow depth strata.)
7. Fish speeds are generated independently according to some common distribution which
does not depend on depth within each stratum.
8. Fish swimming speed is negligible relative to the boat speed. (This assumption will
later be relaxed.)
9. Fish behavior is not altered by the survey vessel. (Although research concerning this
issue is lacking, sockeye salmon likely swim too deep in the turbid waters of the Fraser
River to be very susceptible to boat avoidance. For example, few migrating sockeye
at the study site swim at depths above 3 m (Levy et al. [23]), and few are caught near
the surface in test fishing nets downstream of Mission.)
10. There are few fish swimming close enough to the surface and to the transducer to go
undetected. (Since the large majority of sockeye at the study site migrate a t depths
below 3 m (Levy et al. [23]), and the minimum detection depth is likely between 1.5
and 2.5 m. [R. Kieser, Pacific Biological Station, Nanaimo, B. C., V9R 5K6, pers.
comm.], it is likely that few sockeye salmon were undetected.)
11. The fish swim directly upriver. (A site should be chosen that minimizes opportunities
for fish to mill about.)
12. Fish that cross the beam in a stationary sounding are independently and uniformly
distributed across the beam. (This is essentially equivalent to a key assumption in
Crittenden et al. [12].)
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 2 1
13. The observations from both transect and stationary soundings, though typically sys-
tematically sampled over time, can be viewed as being randomly sampled over time.
(If, for example, there are substantial daily cycles, departures from this assumption
could be minimized by stratifying over shorter time intervals. Failure to do so would
inflate the estimated variance.)
14. The transect and stationary soundings sample fish with the same characteristics such
as speed, size, depth, and orientation. (The validity of this assumption can be pro-
moted by ensuring that the stationary soundings are taken at times that are inter-
spersed spatially and temporally with the transect soundings.)
15. The target widths of individual fish in the stationary soundings are generated inde-
pendently. (This assumption would be violated if, for example, fish speeds displayed
a strong diurnal cycle, and the results were pooled over an entire day. The measured
widths of adjacent targets, coming from roughly the same time of day, would then be
positively correlated.)
3.1.3 The Estimator
Upstream Migration Rate
Suppose the water column is divided into several depth strata within each of which
the number of fish per unit volume does not vary appreciably with depth. First, consider
any depth stratum and assume that target strength is the same for each fish in the depth
stratum.
Let
r = number of fish migrating upstream per second in the depth stratum: the
quantity to be estimated,
cr = width of the river,
zl = depth at the top of the depth stratum,
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 22
z2 = depth at the bottom of the depth stratum,
X = average number of fish per unit volume in the depth stratum: unknown,
and
y = maximum speed of fish (distance per second): an unknown constant.
Next imagine a barrier across the river. At any given moment, which will be referred to
as 'the start', fish that are capable of crossing the barrier within one second from the start
will be within a distance y downstream from the barrier. The volume of the river section
that contains these fish is a(z2 - zl)y and the expected number of fish in this volume is
Xa(z2 - z l ) ~ . However, only a fraction of these fish cross the barrier within one second from
the start, because not all fish travel at the maximum speed 7. Let
,L3 = probability that a fish will cross the barrier within one second from the start.
Then the expected number of fish that will cross the barrier within one second is
r = Xa(z2 - zl)yp.
Calculation of P
Let
D = distance from the fish to the barrier at the start: a random variable,
S = speed of the fish: a random variable, and
B = time taken by the fish to reach the barrier: a random variable.
Suppose that S has some probability density function fs(s) whose domain is [O, y]. Over
the small distance (7) being considered, D will be essentially uniformly distributed in [0, y].
Thus the probability density function of D is fD(d) = $. The variables, D and S , will also
be essentially independent, with joint probability density function,
f ~ , s ( d , s) = f ~ ( d ) fs(s) = fs(s), for d and s both in [0, y]. 7
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 23
Note that B = 9, with B then restricted to the interval, [0,1]. Let U = S. Then D = BU.
The joint probability density function of B and U is
f ~ , ~ ( b , u) = fDYS(bu, u)J, where J = u is the Jacobian of the transformation, 1
= -fs(.).; (b, 4 E A, Y where A is the region with boundaries, b = 0, u = 0, u = and bu = y .
Therefore, the marginal probability density function of B in region [O,1] is
where ps = E(S) is the mean fish speed.
Now,
Substituting for P in formula (3.1) yields
This formula shows that the number of fish passing the barrier per second is equal to
the number of fish per unit upstream distance [Aa(z2 - zl)] times the mean fish speed (p,).
The next step is to calculate the mean fish speed in terms of quantities associated with
stationary soundings.
Calculation of Mean Fish Speed, ps
Consider a stationary sounding. Let Z be the random variable representing the depth
of a fish relative to the transducer. Assume that the horizontal cross section of the beam
at depth z is a circle with radius r(z). Now consider the fish that swim through the beam
at depth 2. Suppose that S is the speed of the fish, C is the distance that a fish swims
through the beam (chord length) and T is the time taken to do so. Speed is distance over
time. However, mean fish speed is not equal to mean distance over mean time. Because C
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
and S are independent, the mean speed can be obtained as
l/E(l) This formula can also be written as p, = ---+. This implies that mean fish speed is ~ / E ( F )
equal to the ratio of harmonic mean chord length to an harmonic mean time.
The next two steps are devoted to obtaining expressions for the numerator and the
denominator of formula (3.3).
Calculation of Mean Reciprocal Time, E($)
Let
W = width of the target in the echogram: a random variable and
p = paper speed of the recorder.
Then, W = Tp. For notational convenience in the subsequent calculations, let M = +. Then,
This formula can also be viewed as 1/E($) = p [ l / ~ ( + ) ] , which means that the har-
monic mean target width is equal to the paper speed times the harmonic mean time.
Calculation of Mean Reciprocal Chord Length, E(&)
Let @ be the half angle that is formed in the center of the horizontal cross section at
depth Z by the chord that a fish travels in the beam (Figure 3.4). Then, C = 2r(Z) sin(@)
and hence,
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
Figure 3.4: The distance that a fish swims to cross through the acoustic beam (C) can be
calculated from the radius T(Z) of the beam at depth z and from the half angle 8 formed in
the center of the beam
Within each depth stratum, fish depths are distributed uniformly (Assumption 6). How-
ever, the sampling probability for a fish at depth z is proportional to T(z). Hence for the
sampled fish, the probability density function fZ(z) of Z is
Let
S = average diameter of the beam = 1; 2~(z)dz. (22 - z1)
(3.7)
Formulae (3.6) and (3.7) lead to
f z ( 4 = ~ ( z ) , for ZI < z 5 z2. S(z2 - ~ 1 )
Then,
Now let fo(8) be the probability density function of 0. It follows from Assumption 9 (that
fish do not respond to the survey vessel) that fish that cross the beam do so along a line
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 26
that is a uniformly distributed distance away from the beam centre. From this, it follows
in turn through a change of variables that fo(8) is proportional t o sin(8). Then,
sin(8) f m = 2 , for 0 5 19 5 n, and
Substituting formulae (3.9) and (3.10) in formula (3.5),
Formula (3.11) can be written as I/E(&) = as. That is, the harmonic mean chord
length is equal to the product of (i) the average beam diameter and (ii) a correction factor,
2/n, which adjusts for the shorter swimming distances for fish that do not cross the centre
of the beam. The factor 2/n is valid for any beam with circular horizontal cross-sections.
Substituting formulae (3.3), (3.4) and (3.11) into formula (3.2) leads to
This formula still contains quantities which are not readily estimable. The next step
shows that the quantity, Xa(z2 - z1)6, is an estimable parameter arising from transect
soundings.
Calculation of Expected Number of Fish per Transect
Let
N = number of fish detected per transect in the depth stratum.
Ignoring river bottom irregularities, edge effects near the shorelines, and fish less than the
minimum detection distance from the transducer, the volume of the depth stratum that is
ensonified by a transect sounding, v, can be approximated by
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 2 7
Now consider an ideal situation where the boat is moving at such a speed that a transect
sounding can be considered a snapshot across the river (Assumption 8). Then, the expected
number of fish that are ensonified during a transect sounding is
However, in practice, there is a maximum feasible speed of the boat. Therefore the assump-
tion of high boat speed introduces a bias to the estimator of E ( N ) and thereby to the final
estimator of r. This is discussed later in Subsection 3.1.4.
Calculation of Fish Passage per Second, T
Now we can combine formulae (3.12) and (3.14) to obtain T , the number of fish migrating
upstream per second, in terms of estimable parameters as
Extension to Unequal Target Strengths
So far, we have assumed that fish target strength is a constant. Now we allow it to be
a random variable and denote it by K. For simplicity, assume that K is a discrete random
variable that takes values in a countable set. The result for continuous K is similar, but the
proof is technically more complex. Let r(k), X(k), v(k) and N(k) refer to the corresponding
quantities above, but for fish of target strength K = k. Then,
N = E N @ ) , and
Analogous to formula (3.14), E[N(k)] = X(k)v(k). Thus formula (3.17) can be written as
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
Let E ( M I K = k) be the conditional expectation of M given K = k. Then,
E ( M ) = E [E(MIK = k)] = C E ( M ~ K = k) W v ( k )
k Ck W v ( k ) '
where is the probability mass function of target strengths of fish Y k ) v ( k )
beam. Substitution of E ( N ) for C k X(k)v(k) in formula (3.18) yields
(3.18)
in the stationary
Analogous to formula (3.15),
Then, formulae (3.19) and (3.20) lead to
which again gives formula (3.16). This shows that even with fish of unequal target strengths,
r can be written in terms of estimable quantities that do not depend on parameters relating
to beam shape or target strengths of fish.
Now we attach subscript i to T, N and M to denote the corresponding quantities for the
ith depth stratum. Finally, summing over all depth strata, we can write the total number
of fish migrating upstream per second, T, as
Estimation of Fish Passage per Second, T
To estimate T, we first need to estimate E(N;)'s and E(M;)'s. We use data from transect
soundings to estimate E(N;)'s and data from stationary soundings to estimate E(M;)'s. Let
N; = sample average of number of targets per transect in the ith depth stratum,
and
M; = sample average of inverse target widths from the ith depth stratum.
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
Then,
can be used as an estimator for T
3.1.4 Bias
Assuming that a transect sounding is a snapshot across the river is equivalent to assuming
that the boat is traveling infinitely faster than the fish. Since the boat speed is limited in
practice, this assumption is false and introduces a bias to the estimate. If the boat speed
is not substantially greater than the fish speed during transect soundings, then the beam
will ensonify fish that are in the sampled volume when the boat arrives, as well as fish that
move into the sampled volume as the boat passes. These latter fish are the source of the
bias. As the beam detects more fish than the model predicts, this bias is positive.
Fish that deviate from the upstream direction when migrating upriver will also contribute
a positive bias. This includes fish that swim downstream, across the flow, or even a few
degrees from the true upstream direction. The fish that do not swim directly upstream
are not moving upstream as rapidly as the method assumes. For fish that swim at an
angle, Q, off the upstream direction, the upstream component to the velocity will be the
fraction, cos(Q), of their overall speed. This does no affect the calculation of average fish
speed using stationary soundings, because of the circular symmetry of the beam. But, by
taking this average fish speed as the average upstream fish speed, the upriver migration rate
is overestimated. However, for Q close to 0•‹, cos(Q) is close to 1. Hence, as long as fish
movement is strongly directed upriver, this bias will not be serious.
Formula (3.15) now becomes
2P T = -XvE(M)E [cos(Q)]
7T
Since Q is small,
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
Suppose that E(@) = 0 and V a r ( @ ) = a;. Then
E [cos(@)] M E [l - m2/2] = 1 - 4 1 2 .
So, (3.23) yields
The relative bias (RB) of 3 is
Since from formula (3.22) 3 for one stratum is i = $NM, the above formula becomes
Recall that N is the average target count per transect sounding, and is based on mobile
soundings, while M is the average of reciprocal target widths and is based on stationary
soundings. These two quantities are therefore independent. Hence, the above equation
simplifies to
Now let
u = speed of the boat.
Then, the time taken for one transect is t = :. Boat speed relative to a fish swim-
ming upstream at speed S is Ju2 + S2 + 2uS sin(@). Therefore, the distance that the
boat would have traveled relative to the fish during time t is t d u 2 + S2 + 2uS sin(@) =
zJu2 + S2 + 2uSsin(@). This corresponds to a volume,
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 3 1
Therefore,
Then, formulae (3.25) and (3.26) lead to
Now, note that
= E [yl] u S2 + u2
- - E [\im\i=]
Therefore, (3.27) leads to
As the bias is positive, the effect of low boat speeds is to overestimate r. However, by
moving the boat at a high speed relative to fish speed, the bias can be reduced. We can
obtain bounds for the relative bias as follows.
From Jensen's inequality (see Appendix) and the fact that 41 + $ is convex in S but
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 32
concave in S2, one can show that
Therefore,
That is,
Hence, a lower bound can be placed on the bias if the mean fish speed and a@ are known.
An upper bound can be added if the variance is also known, and a proper bias adjustment / -\
factor can be computed from an estimate of
For example, assume that a@ was 10 degrees, and the boat speed was close to 1.5 m s-'.
Quinn and terHart [29] report for sockeye salmon (Oncorhynchus nerka) a mean fish speed
of 0.66 m s-l, with a standard deviation of 0.21 m s-'. With these Figures the relative bias
would be between 10% and 12%.
3.1.5 Estimation of Variance
The variance of .i as given in formula (3.22) can be written as
where i and j are stratum numbers. Let 6 = ri + E; and Fj = rj + ~ j . Assume that the
errors E; and ~j are uncorrelated. Then, Cov (Fj,Fj) = COV(E;, ej) = 0, and
var(F) = v a r (6) = [:] v a r (&ai) . i i
Recall that N; and Mj are independent for all i and j.
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
According to Kendall and Stuart [IT],
Var(N;M;) = V~T(N; )V~T(M; ) + v~T(N; )E~(M; ) + v a r ( ~ ; ) ~ ~ ( N , ) . (3.31)
Let
1 = the number of times the boat crossed the river, (3.32)
m; = total number of targets in depth stratum i from the stationary soundings,
s k i = sample variance of the number of targets per transect sounding in depth
stratum i, and
s k i = sample variance of the reciprocals of m; target widths in depth stratum i
from the stationary soundings.
Since % and% are unbiased estimators for ~ a r ( ~ ; ) and v ~ T ( M ~ ) , respectively, ac-
cording to Kendall and Stuart [17] an unbiased estimate of the variance of 7̂ can be obtained
by
3.1.6 Numerical Example
Data from August 31, 1993, are used to demonstrate the methods described in this
section. Table 3.1 contains the summary statistics of the data and estimates of the number
of fish per second in each depth stratum that migrated upriver, along with the estimated
variances. The paper speed of the recorder is 0.478 mm s-' for these calculations.
The estimated number of fish migrating upriver per second is 3. To estimate total
numbers migrating over a time period of duration H , we can use H 3 with estimated variance,
~ ~ ~ a r ( i ) .
The total daily abundance is estimated with H = 86400 s d-' as described above.
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 34
Transect Observations
Depth
Stratum
(i)
1
2
3
Depth
Range
(meters)
0 - 5
5 - 10
10 - 15
#
Passes
( 1 )
Stationary Observations Migration Rates
Ave.
Count
( 4
#
Fish
(mi >
Table 3.1: Summary statistics of the data collected by the Pacific Salmon Commission on
August 31, 1993, at Mission, B.C. Also shown are the estimated numbers of fish migrating
upriver per second and the corresponding estimated variances.
Var.
(sif,)
0.136
0.066
0.130
Var.
( 4 , )
Ave. Recip.
Width
(Mi)
Estimated abundance = HP = H C 6 = 196,998, i=l
Estimated standard error = G(HF) = H JG = H Par(+,) = 8,651, i=l
Est.
(h )
0.39
1.47
0.42
G(H3) Estimated coefficient of variation = -% = 4%.
H P
Var .
[ P a w ]
0.0107
0.0815
0.0164
Fish abundance on this day (about 197,000 fish) was about half the largest daily abundance
(about 403,000 fish) observed in 1993.
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
3.2 Estimation of Resident Fish Populations
In this section we adapt the ideas developed in Section 3.1 to derive an estimator for the
total number of fish in a resident population; for example, in a lake. For this derivation,
we make the assumptions 1-10 stated in Subsection 3.1.2, and in addition, the following
assumptions.
11'. The fish swim around in random directions at slow speeds.
12'. The fish that are crossed by the moving beam are independently and uniformly dis-
tributed across the beam.
13'. The survey area is randomly sampled by transect soundings.
Here, we are not estimating the numbers of fish that pass a given point. Hence, one main
difference of this situation from a migrating fish population in a river is that calculation of
fish speed is not necessary. For this reason, stationary soundings no longer provide relevant
information.
3.2.1 The Estimator
As in Subsection 3.1.3, suppose that the water body is divided into several depth strata
within each of which the number of fish per unit volume does not vary appreciably with
depth. First, we estimate the number of fish in a given depth stratum assuming that all the
fish in the stratum have the same target strength. Later we relax this assumption. Then,
the total number of fish is estimated by adding the stratum estimates.
Let
T = total number of fish in the depth stratum: the quantity to be estimated,
A = survey area: assumed known,
zl = depth at the top of the depth stratum,
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
depth at the bottom of the depth stratum,
total volume of the water in the stratum: assumed known,
number of fish per unit volume in the depth stratum: unknown,
boat speed: assumed known,
length of a transect: assumed known,
expected number of fish to be ensonified in the depth stratum
during a transect :unknown,
observed number of fish ensonified in the depth stratum
during a transect, and
effective volume sampled in a transect: unknown.
By the definition of A,
Under the random sampling assumption,
Since vo = A(an - zl), (3.34) and (3.35) lead to
Now let 6 be the average diameter of the effective beam defined by (3.7). Since v = a(a2 -
z1)6 by (3.13), T can be written as
From (3.11),
Chapter 3. Es tirna tion of Fish Populations Using Echo Sounding Techniques 3 7
In (3.11), C was the distance that a fish had to swim to cross a stationary beam. This
formula is still valid for our purpose, but now C is the distance that the boat must travel
for the beam to pass over the fish.
Now let
v = the pulse rate of the transducer,
e = the number of echoes received from a fish, and,
e, = l l e .
Then, according to Crittenden et al. [12],
Using equations (3.36), (3.37) and (3 .38) , T can be written in term of estimable quantities
as
Extension to Unequal Target Strengths
Now suppose that the target strength of a fish is a random variable K . For simplicity,
assume that K is discrete that takes values in a countable set. Let n ( k ) and v ( k ) refer to
the corresponding quantities above, but for fish of target strength K = k . Then,
and the probability mass function p ( K = k ) of fish of target strength k that are ensonified
in the beam is
Let E(e , lK = k ) be the conditional expectation of e , given K = k . Then,
n ( k ) E(e , ) = E [E(e , lK = k ) ] = E(e, lK = k ) - , k
n
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
from which we get
Furthermore, according to (3.39),
2Av T(k) = -n(k)E(e,lK = k).
Tau
By summing (3.44) over k, we obtain the total population size under the new assumption
that target strength is a random variable. Denote this total by TI. Then,
which is again equal to T given by (3.39). This implies that the population size can be
written in terms of estimable quantities which are independent of the target strength dis-
tribution.
Estimation of T
Suppose that 1 transects are randomly sampled. For each transect, the number of fish
ensonified is counted and the length of the transect is recorded. The number of fish per unit
distance is then calculated. Also, the number of echoes per fish is determined for all fish
observed or a random sample of them. Let
N = average number of fish observed per unit distance and
e, = average reciprocal number of echoes per fish.
The total number of fish in stratum, T can be estimated by
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
The total number of fish in the survey area can be estimated by adding the stratum
estimates over the depth strata.
Note:
For a paper recorder, we can write e , in terms of the paper speed of the recorder and the
mean reciprocal target widths. Denoting the parer speed by p and the target widths by W,
it is easy to see that W = p(e/v). Hence, e , = l / e = p/(vW) = (p/v)M, where M is the
reciprocal target width. Therefore, (3.39) can also be written as
Consequently, T can be estimated by
where M is the sample average of reciprocal target widths. Note that this estimate has the
same form as (3.22) except for the multiplicative constant.
3.2.2 Estimation of Variance
It is reasonable to assume that N and M for a given depth stratum are independent. Also,
the total estimates in different strata are uncorrelated. Hence, similarly to (3.33), the
variance of the total estimator can be estimated by
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques
3.3 Discussion
The described method for estimating the number of migrating fish in a river using hy-
droacoustic techniques has two major advantages over similar methods used in lakes and
reservoirs. First, the method is not sensitive to beam shape or dimensions. Second, the
exact calculation of the expected reciprocal of the chord length allows the user to avoid the
use of the bootstrap to reduce bias caused by using an estimation formula with a random
variable in its denominator.
We identified several potential sources of bias inherent in the method. The first, which
stems from the speed of the boat being only moderately faster than fish migration speed,
tends to result in overestimation of abundance. This bias can be reduced by increasing the
boat's speed as much as practically possible, or by estimating a bias adjustment factor that
can be extracted from formula (27).
Second, fish swimming off the upstream direction will introduce a positive bias. The
stationary soundings estimate overall fish speed, and the upstream component to the velocity
will be overestimated. This source of bias can sometimes be reduced by choosing a location,
such as the Mission site, where there are limited opportunities for the fish to mill about.
Contrasting with these sources of positive bias are sources of negative bias. Fish that
travel in areas of the river that cannot be acoustically sampled, such as close to the surface
above the minimum detection depth and close to shore where the echo-sounding boat cannot
travel, will be missed. Similarly, fish that sense and avoid the echo-sounding boat will lead
to a negative bias. Finally, when the targets of two or more fish overlap on the echogram,
so that only one target is apparent, the population will be underestimated. These biases
are not considered to be significant for sockeye salmon in the Fraser River.
In determining depth strata, we have taken the distances determined by the echosounder
as actual depths. This can also introduce a bias. This bias can be reduced by reducing the
beam angle. Ways of reducing bias are also being investigated through the application of
new technology (e.g., split beam).
Chapter 3. Estimation of Fish Populations Using Echo Sounding Techniques 4 1
The estimated variance for the worked example could possibly be reduced further if the
day was divided into smaller units of time. This is because a daily cyclical pattern in the
abundance of migrating sockeye salmon is ignored in our example. Stratifying each day such
that the strata are homogeneous with respect to migration (e.g., into hourly strata) could
result in more precise estimates. Daily abundance and variance can then be estimated as
the sum of the stratum estimates.
The fish speed calculated through target widths obtained from stationary soundings
plays an important role in the estimation. Fish speed can vary substantially over time
and space (depth and location in the river). Therefore, the timing and location of sta-
tionary soundings should be re-examined, and alterations to the sampling design should be
considered to address concerns about possible biases.
In the example shown, the 264 targets sampled in the stationary soundings provided
estimates with reasonable precision. On days with fewer fish in the river, the estimates will
have a larger coefficient of variation. A scheme for optimal allocation of sampling effort
between the two types of soundings could substantially reduce the coefficient of variation,
particularly on such days.
The estimators are sums of products of averages, and may have skewed distributions.
Therefore, the usual confidence intervals, calculated by adding and subtracting standard
errors times normal quantiles may not provide an accurate coverage. In Chapter 4, we
investigate two alternative strategies for this purpose.
The method described in this chapter has a strong potential to give accurate and precise
estimates of fish passage that are robust and comparatively easy to obtain. Methods for
promoting and checking the validity of the underlying assumptions are discussed. The
method provides a significant tool for managers under pressure to make timely and effective
management decisions.
Chapter 4
Confidence Intervals for a Sum of
Pairwise Products of Means
The purpose of this chapter is to develop methods to find confidence intervals for a sum of
products of means. The motivation arose from the requirement to find confidence intervals
for the migration rate estimated in Section 3.1 and the number of fish in a resident population
in Section 3.2. Both estimators are sums of products of pairwise means. Products of means
also arise in other practical situations. As Bergeret al. [5] pointed out, they arise in
determining area based on measurements of length and width. Furthermore, in gypsy moth
studies, the hatching rate of larvae per unit area can be estimated as a product of the mean
number of egg masses per unit area times the mean number of larvae hatching area per egg
mass. Applications also arise in forest sampling (A. Ellingsen, personal communication).
In the fish applications described here, the independence of the individual means follows
naturally from the assumptions about the sampling procedure. In other applications also
approximately independent samples can be obtained for each mean (Southwood [37]).
Berger et al. [5] considered the problem of obtaining confidence intervals for a product
of two means from independent normal distributions with known variances when only one
observation from each distribution is available. They proposed a Bayesian approach with
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
reference priors. We consider the problem assuming that large samples are available from
each distribution, and explore two methods that are derived from the likelihood ratio test
and the Wald test. In all the applications described above, the variables involved are
positive. For this reason, in our development we assume that the variables are positive.
Possible modifications for other cases are also suggested.
In the area-estimation example, both variables, the length and the width, may be well-
approximated by normal distributions. In other applications, one or both variables can
be far from normal. In the fish problem, one variable is the number of fish detected per
transect. The other variable in this case is the reciprocal target width of a fish on an
echogram. The latter is proportional to fish speed. Figure 4.4 shows histograms of these
variables based on data collected by the Pacific Salmon Commission on August 31, 1993 at
Mission, B.C.. According to thesehistograms, the variables seem to have positively skewed
distributions. However, when the samples are large, sample averages are approximately
normally distributed. Hence, the methods used to construct confidence intervals for sum of
products of normal means may be adapted in these situations as well.
So, we first develop methods for finding confidence intervals for a sum of pairwise prod-
ucts of means from independent normal distributions. These methods are based on the
standard, general likelihood ratio and Wald tests. The methods are then extended to other
distributions using the central limit theorem. In the case of normal means, Bartlett-type
adjustments are derived to improve the confidence intervals obtained through the Wald test.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
4.1 Notation and Assumptions
4.1.1 Notation
Let p be the number of pairs of products in the sum. When we have only one pair of
products ( p = I) , we use the following notation:
1 for the first component of the pair
2 for the second component of the pair
pj = the meanof the jthdistribution,
u: = the variance of the j t h distribution,
nj = size of the sample from the j t h distribution,
j = the sample average from the j t h distribution,
To further simplify notation, also let
% = (::), v = (::), and + = (::).
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 45
When p > 1, we attach an additional subscript to denote the pair. For example, for
i = 1,2, . . . ,p ,
j = the mean of the j th distribution of the ith pair,
2 aij = the variance of the j th distribution of the ith pair,
n,j = size of the sample from the j th distribution of the ith pair, and
j = the sample average from the j th distribution of the i th pair, etc..
In addition, let
n o = the smallest sample size P
P = C~lPi2, k l
S = -2 times the maximum log-likelihood ratio [as defined by (4.2)], and
W = the Wald statistic [as defined by (4.30)] .
4.1.2 Assumptions
The only main assumptions we carry throughout the discussion are,
1. all the variables involved are independent of each other, and
2. all the observations within a sample are independently, identically distributed.
4.2 Normally Distributed Data
4.2.1 Likelihood Ratio Approach
One approach is to construct approximate confidence intervals by inverting the likelihood
ratio test. Consider the null hypothesis, H o : p = po. The likelihood ratio for testing this
hypothesis is
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 46
where 6(8; x ) is the likelihood function. According to Wilks [41] , -2 times the log-likelihood
ratio,
S ( p o ) = -2 In W p o ) ( 4 . 2 )
has a limiting chi-square distribution under Ho. The method proceeds by determining the
likelihood ratio test of the null hypothesis for a specified value of po, and then inverting this
test in the usual way to obtain a confidence interval for p as those values which do not lead
to the rejection of the null hypothesis.
Product of Two Normal Means: Variances Known
Let us first consider the case of two independent normal means with known variances. The
likelihood ratio for testing Ho : p = po is
where
is the likelihood function. It is easy to see that the logarithm of the denominator of L R ( p o )
is
To determine the numerator of L R ( p o ) , we need to find the maximum of 1(p; x ) subject
to the constraint p1p2 - PO = 0. This is equivalent to finding the maximum of In I(p; x )
subject to p1p2 - PO = 0. We use the Lagrangian multiplier method to find the restricted
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
maximum likelihood estimates. Let
where X is a Lagrange multiplier. The critical point of h(p, A) is found by solving the
equations,
a h - - - 31 - P1 -- X p 2 = 0 , and dPl u1
The critical point thus obtained is
where
z1 - X U ~ ~ ~ X X ) = 1 - X2ulu2 , and
such that X satisfies the constraint equation,
= Po.
Note that b(O) is the unrestricted maximum likelihood estimate. The constraint (4.11)
reduces to a fourth degree polynomial and gives up to four solutions for A. Since we assume
that both variables are positive, jil and ji2 should also be positive. Therefore, some solutions
can be eliminated. From the other solutions, we should choose the one corresponding to
the maximum of h(X). Then, substituting these values in (4.2), S(po) can be calculated. If
S(po) is less than or equal to X:,(l-,,, then po is included in the confidence interval. The
procedure is systematically repeated for different po to construct the confidence interval.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 48
However, this can be tedious because we have to solve (4.11) for each po. Madansky [25]
introduced a method which simplifies these computations. This method avoids the need to
solve (4.11) for different values of po. Instead, it obtains the restricted maximum likelihood
estimates in terms of the Lagrangian multiplier, X [as in (4.9) and (4.9)]. Then, substituting
these in (4.2), S is obtained in terms of A. Let this be denoted by S(X). Madansky's method
solves the equation,
for A. Usually, this equation has two real roots, and can be solved easily if S(X) behaves
well around the unrestricted maximum likelihood estimate. The roots are then substituted
in the constraint function (4.11) to get confidence limits for p = plpz. The following result
due to Madansky [25], provides a necessary and sufficient condition for S(X) to have such a
behavior around zero (zero is the value of X that corresponds to the unrestricted maximum
likelihood estimate of p) .
Theorem 1 (Madansky) -2 times the logarithm of the likelihood ratio is a monotonically
decreasang (increasing) function of X for X < 0 (A > 0) if and only if the constraint function,
as a function of A, is monotonically decreasing.
When the likelihood ratio and the constraint function exhibit such behavior, it is easy to
invert the likelihood ratio test to construct confidence intervals as described above. We now
investigate the possibility of using Madansky's method for our problem. Figure 4.1 shows
an example of the behavior of p(X) and S(X) when ZIZz > 0. This example was constructed
using nl = 30, n2 = 30,Zl = 10, Z2 = 20, sl = 3 and s 2 = 5. Note that p(X) is positive and
monotonically decreasing in an interval containing 0, and in this interval, S(X) is convex.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 49
0
LAMBDA
LAMBDA
Figure 4.1: Behavior of p(X) and S(A) when f t l f t 2 > 0 (example calculation using nl =
30, n2 = 30, 51 = 10, Z2 = 20, sl = 3 and s2 = 5).
The following result identifies this interval in general, and shows that (4.11) has a unique
solution Xo in this interval such that pl (Xo) and j i 2 ( X O ) are positive. Further, it shows that
j i ( X o ) is in fact a local maximum of 1 subject to the constraint, h1(X)h2(X) - po = 0. So,
the unique local maximum of the likelihood subject to the constraint can be easily found.
In other words, this result enables us to pick up the unique set of local maxima that gives
us a continuous relationship between fi and p for p near po.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Result 4.1 Let
Suppose that il > 0, and Z2 > 0. Then,
(a) p ( X ) is monotonically decreasing in IA, and S (X) is monotonically decreasing (increas-
ing) for X < 0 ( A > 0 ) in I A .
(b) jl is positive in JA and (4.1 1) has a unique solution X o in Jx.
(c) p = f i ( X O ) is a local maximum of I(p; x) subject to the constraint p1p2 - PO = 0.
Proof
( a ) Note that
Let
To prove part (a ) , it is sufficient to show that the derivative of g with respect to X is negative
in Ix. This derivative is
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 5 1
Now suppose without loss of generality that 71 < 7 2 . Then it is clear that $ < 0 for
X E (-PI, 0). To show that $f is negative for A E [0, PI), we simplify (4.13 ) obtaining
Since the denominator of (4.14) is negative for X E [0, PI), we need to show that the numer-
ator,
of (4.14) is positive in this interval. We show this by negating a contradiction. Suppose
that q(X) is non-positive for some X E [0, PI). Now, note that q(X) is differentiable in [0, PI).
Also note that
So, q(X) must have a non-positive minimum in [O,P1). In order to examine the critical
points of q(X), consider the derivative
The derivative is zero at rl and 72. But, q(rl) = 71 (71 - 72)2 > 0, and P1 < 7 2 (since
yl < fl < 72). Therefore, q(X) does not have a non-positive minimum in [0, PI). This
proves that q(X) is positive in [0, PI) and conseqently that p(X) is monotonicaJly decreasing
in I x . Then, by Madansky's theorem, S(X) is monotonically decreasing (increasing) for
X < 0 (A >O)in I x .
( b ) The denominators in (4.9) and (4.10) are positive as long as [XI < J-, and
numerators are positive as long as X < min(yl , 72). This implies that fi is positive in J x .
Now notice that p(X) approaches infinity as X approaches -PI and p(P2) = 0. Therefore,
for any po > 0, (4.11) has a unique real root Xo in JA.
(c) We have already established that fi(Xo) is a critical point of h. According to Marsden
and Tromba [26](pg. 274), in order to prove that fi(Xo) is a maximum, it is enough to show
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 52
that the value of the bordered Hessian determinant of h (see Appendix), when evaluated at
fi(Xo) is positive. The bordered Hessian determinant of h is
B H b , X ) = - p 1 Xp2- - -p2 X p l - - . ( 3 ( t:) Let BHo = BH(ji,Xo). Then, using (4.9) and (4.10),
Since x, u and q(X) are positive in Jx (which is a subset of Ix), BHo > 0 in JA. This proves
the result.
According to Result 4.1, estimates (4.9) and (4.10) do indeed specify restricted maxima.
Therefore, substituting these values in (4.4), the logarithm of the numerator of l(po) can be
written as
Hence, twice the log-likelihood ratio is
Result 4.1 confirms that S(X) monotonically decreases (increases) for X < 0 ( A > 0) in
IA while p(X) monotonically decreases in the interval. Therefore, S can be easily inverted
to calculate an approximate (1 - a)100% confidence interval for p. Note that S(0) = 0
and S(X) approaches infinity when X approaches PI. Since S(X) is continuous in J A , there
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
exist a Xi(< 0) such that S(X;) = x:,(,-,). If S(P2) 2 x:(l), then there also exist a
A;(> 0) such that S(X;) = X:,(l-,). Then the confidence interval is given by [p(X;),/~(Xi)l.
If S(&) < X:,(l-,), there does not exist a X i ( > 0) such that S(Xj) = X:,(l-,). Then the
natural lower bound p(P2) = 0 serves as the lower limit of the interval.
Remark
We have not imposed any sign restrictions on the means in the maximization procedure.
Hence, the method can be used regardless of the sign of the averages. By an argument similar
to that in the proof of Result 4.1, one can show that p(X) is monotonically decreasing in Ix
for all cases. Then, by Madansky's theorem, S(X) behaves well in Ix. So, S can be inverted
easily to construct the confidence intervals.
Sum of p Products of pairwise Normal Means: Variances Known
Now we extend the method to construct confidence intervals for the sum of products,
Here, (pil , pi2) is the pair of means of the ith product. By direct extension of Result 4.1,
it can be proved that the following result is true.
Result 4.2 Let
u;1 =
uj2 =
Yil =
Yi2 =
P1 =
IA =
S(A) =
min [,/-I = , min i=l,..,p z=l,..,p
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
such that X satisfies
Then,
1. p(X) is monotonically decreasing in I A , and S ( X ) is a monotonically decreasing (in-
creasing) for X < 0 ( A > 0 ) in I A .
2. Under the null hypothesis Ho : c:=, pjlpi2 = po, the S is asymptotically distributed
as xq.
Therefore, an approximate ( 1 - a)100% level confidence interval can be calculated using
Madansky's approach.
Product of Two Normal Means: Variances Unknown
Usually, in practical situations, the variances are unknown. We now develop likelihood ratio
methods for the case of normal data with unknown variances.
The likelihood function (4.4) should now be viewed as a function of both p and u. The
log-likelihood function is
Method 1
It is very easy to find the unrestricted maximum likelihood. But it is not so easy to find the
maximum subject to the constraint plp2 - po = 0 . The critical points of the Lagrangian
h = In 1(p, u; x ) - X (p1p2 - pO) can be found by solving the system of equations,
Chapter 4. Confidence Intervals for a Sum o f Pairwise Products o f Means
ah -- - -- nl s: + (31 - P I ) 2 + 2u:
= 0, and dul 2u1
The Maple V solution of this system o f equations can be simplified as follows:
b2(X) = a root o f the equation, csPg + ~ 4 ~ ; + C ~ P ; + C Z P ; + C l P 2 + CO = 0 , (4.24)
where
co = - (sz2 + q2) nl [ -22 n2 + " 1 ( s z 2 + ~ 2 ~ ) A] ,
ci = c12A2 + C I I A + clo,
with ell = (s12 + 3;) (s12 + 3:) ,
ell = 2 32 (s22 + 3;) (2 nl - n 2 ) ,
C I O = -n2 (-3;n2 + nl sz2 + 3 nl 3;) ,
c2 = c22x2 + c21x + c20,
with c22 = -4 32 ( s z2 + 3:) (s12 + 3;) ,
c2l = -2 it1 (3 3; + s12) (nl - n l ) ,
c20 = 32 n2 (-2 722 + 3 721) ,
c3 = c32x2 + c31x + c30,
with c32 = 2 (3 3; + (s12 + 3:) ,
cgl = 2 3 2 ( 2 n1 - 3 n 2 ) ,
C 3 0 = -n2 (n l - n2) ,
c4 = c42x2+c41x,
with ~ 4 2 = -422 s1 + xl , ( - 2 )
c41 = -31 (n l - 2 n 2 ) , and
c5 = ( s l 2 + s : ) x 2 ,
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
and
where X satisfies p(X) = ji1(A)b2(A) = PO.
In general, an algebraic solution of (4.24) is not possible. So it has to be solved nu-
merically. This can easily be done with IMSL subroutine 'zplrc'. If more than one root is
real, we have to choose the root that corresponds to the maximum of the likelihood. In the
present case, we can ignore all the negative roots because we assume that the means are
positive. We found that for negative A, bl(X) is greater than 31 and b2(X) is greater than
22. Also, for positive A, both estimates are smaller than the corresponding sample averages.
Obviously, the case X = 0 corresponds to the unrestricted maximum likelihood. We found
that p(X) is monotonically decreasing in an interval including the origin. Therefore it is easy
to invert the likelihood ratio using Madansky's approach to find confidence intervals.
This procedure has the advantage that it can be extended for a sum of products of
means. However as the p2 has to be estimated numerically, and the valid root has to be
selected from five roots, this procedure can be computationally intensive. We now suggest
another method which is computationally more attractive, but can be used to construct
confidence intervals for one positive product of means only.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products o f Means 5 7
Method 2
Suppose that p = pip2 > 0. We can now formulate our restriction as 1n(Plp2) = ln(po).
Hence, the critical points of the Lagrangian,
can be found by solving the following set of equations:
nl s12 + (31 - ~ 1 1 ) ~ -- + = 0 , and 2 "'1 2 u12
The MAPLE V solution is as follows. f i l ( X ) and f i 2 ( X ) are roots of equations,
( n i + X)p i2 - ( nl + 2 X ) 1 1 ~ 1 + (s12 + 3:) A = 0 , and (4.25)
( n 2 + ~ ) ~ 2 ~ - ( n 2 + 2 ~ ) 3 2 p 2 + ( ~ 2 ~ + 3 ~ ) ~ = 0 , (4.26)
respectively. The estimates of variances are
fil(A) = s12 + (31 - i q 2 , and (4.27) n1
f i 2 ( X ) = ~2~ + (32 - f i2 )2 (4.28)
n2
Considering the facts that f i l (0) = il and ji2(0) = 32, valid roots can be chosen from (4.25)
and (4.26) as
f i l (A) =
f i l ( -n l ) =
f i 2 ( X ) =
fi2(-712) =
31 ( nl+ 2 X ) + \ l ~ : n l 2 - 4 nl s12 A - 4s12 ~2
2 ( n 1 + A) , for X # - n l ,
x n 2 - 4 n 2 s 2 2 X - 4 s 2 2 X 2 3 2 ( n 2 + 2 ~ ) + J-f 2 , for X # -n2, and
2 ( n 2 + A)
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
where X satisfies jil(X)ji2(X) = PO. TO ensure real roots, the appropriate interval for X is
IA = [Il, 121, where
We can show that jil(X) is non-increasing in Ix. To see this, consider the derivative,
The denominator
To end this, let
is positive in IA. We can show that the numerator is non-positive in IA.
??nl2 + 2 nl x s12 + 2 n12s12, and
(z;n12 - 4 nl x s12 - 4 ~ ~ s ~ ~ ) 2;nl2 - (z;?n12 + 2 nl x s12 + 2 n12s12l2
-4 n12s12 (A + nll2 (s12 + 3:) .
Suppose that both averages are positive (If both averages are negative, all the calcula-
tions can be done neglecting the signs). Then, a is positive in IA. When A > - F ( 2 + $ $), b is positive. Since Il > -7 (2 + $), it implies that b is positive in IA. So, the fact that Q
is negative in IA except a t -nl, proves that (4.29) is negative in IA. This in turn shows that
bl(X) is monotonically decreasing in Ix except a t -nl , at which it has an inflection point.
Similarly b2(X) is also monotonically decreasing in Ix. Consequently, p(X) = ji1(X)ji2(X) is
monotonically decreasing in Ix. Hence, again the likelihood ratio test may be inverted using
Madansky's approach to construct confidence intervals for p. It is possible that there does
not exist a A;(> 0) at which the likelihood ratio is equal to X$l-,). For example, this can
happen when I2 is very close to zero, or in other words, when at least one of the coefficients
of variation is very high (see Figure 4.2). If this happens, p(12) may be taken as the lower
confidence limit.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-20 -1 5 -1 0 -5 0 LAMBDA
Figure 4.2: An example where S(X;) < X:,(l-,): (example calculation using nl = 30,nl =
30,Z1 = 10, S2 = 20,sl = 15, s2 = 5 and cr = .05. Vertical dotted line represents X = 12.
Horizontal dotted line represents x:,,~~.)
Sum of p Products of Pairwise Normal Means: Variances Unknown
Method 1 described above can be extended to construct confidence intervals for a sum of p
products of pairwise means. The method proceeds by calculating the restricted maximum
likelihood estimates of one of the means in each pair using the formulae provided.
As mentioned earlier, Method 2 cannot be extended for sums of products of means.
However, it can be easily extended to construct confidence intervals for a product of q(> 2)
means. This could be especially useful in situations where volumes (q = 3) have to be
estimated.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
4.2.2 Wald Test Approach
Another way of constructing confidence intervals is to invert the Wald test. According to
Silvey [35](pp. 115-117 ), the hypothesis Ho : g(8) = 0, can be tested by calculating the
unrestricted maximum likelihood estimate 9 of 8 and basing our decision on the proximity
to zero of g(9). This method has the advantage that we do not need to calculate the
restricted maximum likelihood estimates. The general
(1943). Accordingly, the Wald statistic,
idea was first exploited by Wald
(4.30)
has a limiting chi-square distribution with one degree of freedom when Ho is true. Here,
the prime denotes the transpose and I is the information matrix.
When the variances are known, for the one product case,
The information matrix is
and the derivative of g is
Therefore, the Wald statistic in this case is
Because of the simple form of this statistic, = , can be easily used to
construct confidence intervals. Since f l has a limiting standard normal distribution under
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Ho, an approximate (1 - a)100% confidence interval can be calculated as
f l f 2 f za12\Iu1E~ + u2f?
In general, for the p product case, the interval is
Similarly, when the variances are unknown, the corresponding Wald statistic leads to the
approximate confidence interval,
Remark
The usual confidence interval obtained by adding and subtracting a multiple of the standard
error is
For large samples, the intervals (4.33) and (4.34) are almost the same because v;j and G;j
are asymptotically equivalent and the terms, GilG,2, are of smaller order.
Bartlett-Type Adjustments
Bartlett [4] introduced the idea of adjustment factors t o improve the chi-squared approxi-
mation to the null hypothesis distribution of the log-likelihood ratio statistic. Suppose that
under the null hypothesis, the expected value of twice the log-likelihood ratio statistic S is
where either b is known or can be estimated consistently. Then,
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 62
has an expected value closer to that of X 2 than has S. Bartlett showed that for the test of
homogeneity of variances, the first three moments of 3 agree with those of Xi with error
~ ( n - ~ l ~ ) , giving strong grounds for thinking that the density of 3 is better approximated
by the X j density. Barndorff-Nielsen and Cox [3] gave a proof of this useful result, namely
that 3 has the xi distribution with error ~ ( n - ~ / ~ ) .
It may be possible to use this idea to improve the confidence interval (4.33) by calculating
an adjustment factor. Since the Wald statistic is asymptotically equivalent to the likelihood
ratio, it is reasonable to expect that a Bartlett-type adjustment will improve the confidence
intervals calculated through the Wald statistic as well.
Consider the statistic,
obtained by respectively replacing v ; ~ and vj2 by 5il and 5;2 in the Wald statistic corre-
sponding to the interval (4.33). Because the 6's are unbiased estimators of the u's, the
expected values of many terms in the Taylor expansion of w become zero. This simplifies
the calculation of adjustment factors. Noting that both W and w have the same asymp-
totic distribution under the null hypothesis, Ho : C:=l p;lp;2 = po, we suggest the use of
w instead of W to calculate the adjustments.
We calculated the derivatives using Maple V functions. The first order derivatives,
when evaluated at O1 are zero. The second order derivatives are O(1) and contribute to
the approximate distribution of W . The expected value of these terms is 1. The third
order derivatives, when evaluated at 9 are zero. We found that in order to obtain the
terms of order l /no2, w should be expanded in a Taylor expansion up to fourth order
derivatives. The higher order derivatives are ~ ( n ; ~ ' ~ ) . The following Conjecture condenses
the derivatives produced by Maple V and suggests an adjustment factor to W.
-
'0 is the vector of all parameters defined as in Subsection 4.1.1 2no = min(n l l ,n~z , . . . , npl,npz)
Chapter 4 . Confidence Intervals for a Sum o f Pairwise Products o f Means
Conjecture 4.1 Let
Then, the fourth order derivatives of W , evaluated at 8, are
a4 w (e) = 0. aqm
The fourth order expected values are
E [ ( ~ i l - pil)4] = 3u:', and
Let
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
c = ( I + A + B ) - l ,
C = estimated C obtained by replacing means and variances by the corresponding sample
averages and unbiased sample variances, and (4.38)
no = . min(n11, n12,. a , npl, np2). (4.39)
Then,
-312 1. E(W) = 1 + A + B + O ( n o ), and
- -312) 2. w = cw has an approzimate X: distribution with error O(no .
So, the confidence interval,
is expected to perform better than (4.33). Some numerical comparisons are presented in
Section 4.4.
4.3 Adaptation to Distributions other than Normal
Now, we address the problem for non normal data with the particular application of the fish
problem in mind. Recall that in this problem, one variable is the reciprocal target width
(M) of a fish on an echogram, which is proportional to the fish speed (stem and leaf plots of
target widths W, are shown in Figure 4.3). The other variable is the number of fish detected
per transect (N). Figure 4.4 shows histograms of these variables based on data collected by
the Pacific Salmon Commission on August 31, 1993 at Mission, B.C.. According to these
histograms, the variables seem to have positively skewed distributions. In the following
table, we have reproduced the summary data in Table 4.1.
One way to construct confidence intervals in this case is to model the data by suitable
probability distributions and then to invert the likelihood ratio in the usual way. But this
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Varmblc = W, No. oC obscrv~~ions = 77 Mcdian = 1 QuarLilcs = 0.8, 1.3
Dccimrl poinL is aL L ~ I C colon
0 : 66 0 : GGGGGGGGGGG777 0 : 000000000000090000000 1 : OOOlililllllli 1 : 223333333 1 : 4444444666 1 : GGGGG 1 : o 2 : o
Variablc = W2 No. or obscrvaLio~~s = 145 Mcdian = 1.5 Qusrlilcs = 1.1, 1.8
Dccinlal poinL is 3L L I I C colon
vaiiab~c = wJ No. of obscrvaLions = 42 Mcdian = 1.45 Quarlilcs = 1, 1.0
Figure 4.3: Stem and leaf plots of target widths, Wi9s (based on hydroacoustic data collected
by the Pacific Salmon Commission on August 31, 1993 at Mission, B.C.).
Table 4.1: Summary statistics of the data collected by the Pacific Salmon Commission on
August 31, 1993, a t Mission, B.C.
Depth stratum ( i ) n; N s ~ , CV(N)% m; M SM, CV(M)%
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Figure 4.4: Histograms of N;'s and M;'s (based on hydroacoustic data collected by the
Pacific Salmon Commission on August 31, 1993 at Mission, B.C.). The height of each bar
is the proportion of observations in the corresponding interval on the horizontal axis.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 6 7
method requires a parametric assumption which may be difficult to justify. We recommend
the following alternative.
With large samples3, by the central limit theorem, we can.expect the sample averages to
be fairly normal. Also, the unknown variances can be estimated with high accuracy. So, the
unbiased estimates of variances may be treated as true population variances. Therefore, the
methods developed for normal data with known variances may be adapted for this problem.
First we adapt the likelihood ratio methods. Now, the maximum likelihood estimates of
variances cannot be derived. Hence exact likelihood ratios cannot be calculated. However,
an approximate likelihood ratio may be calculated using the the approximate likelihood
function,
It is easy to see that this (approximate) likelihood is proportional to that given by (4.4).
Therefore, when the samples are large enough, the likelihood ratio methods developed for
the case of known variances may still be used.
Similarly, approximate confidence intervals may also be calculated using (4.33). For
samples of moderate sizes, one may replace the normal quantiles by t quantiles. As a
conservative approach, the size of the smallest sample may be taken as the corresponding
degrees of freedom. Some comparisons and further discussion are presented in Section 4.4.
4.4 Worked Example and Monte Carlo Studies
4.4.1 Worked Example
To illustrate the methods suggested for constructing confidence intervals in the case of
non normal means, we constructed approximate 95% confidence intervals for the migration
rate for the Fraser River fish problem. We first constructed confidence intervals for p =
3This is usually the case in the Fraser River hydroacoustic application. See Table 4.1.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
c:=~ E(Nj)E(Mi) using statistics S given by (4.22).
(a) -2 times the log likelihood ratio
0 5 LAMBDA
(b) The sum of pairwise products of means
0 5 LAMBDA
Figure 4.5: Calculation of 95% confidence interval for p = c:=~ E(N;)E(M;).
To explain the calculation of confidence intervals through S , let Z1 = N, Z2 = M , s; =
(s&,), S; = (sk,) , nl = n and n2 = m. Treating the sample variances as true population
variances, let u; = s;/ni. The resulting S(X) and p(X) as given by (4.22) and (4.23), and
the calculation of confidence intervals are schematically shown in Figure 4.5. The dotted
horizontal line in (a) represents the value x ~ ~ ~ , ~ = 3.841459. From this figure, it is clear that
S(X) cuts this line at two values between -10 and 10. Using the bisecton method, we found
the corresponding X values to be X = -6.242009 and 6.678391. Substituting them in p(X), an
approximate 95% confidence interval for p was calculated. Then, by multiplying the limits
by x 86,400, a confidence interval for the total abundance can be calculated. We also
calculated confidence intervals for those quantities through the Wald statistic with normal
and t quantiles. These confidence intervals are presented in Table 4.2. In this Table, W,
and Wt denote the confidence intervals obtained with normal and t quantiles respectively.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
95% confidence intervals
Table 4.2: 95% confidence intervals for the fish problem
P = E L E ( M ; ) E ( N ; )
Daily abundance
4.4.2 Monte Carlo Studies
Next we describe several Monte Carlo studies that were performed to assess the appropri-
ateness of the suggested methods.
S
(6.92, 8.11)
(181,940, 213,227)
First, to assess the appropriateness of the methods suggested for non normal data, we
generated large samples from different distributions with the means and variances that are
equal to sample averages and sample variances in the above example (in the case of Poisson,
only the mean can be matched). Treating the sample variances as true population variances,
we conducted 10,000 Monte Carlo simulations. For each simulation, we calculated confidence
intervals based on S given by (4.22) and W given by (4.33). Since the usual confidence
intervals calculated by adding and subtracting the standard errors are almost the same as
those calculated using W (traditional intervals are slightly narrower; see (4.34)), in this
way we can compare the the likelihood ratio intervals and the traditional intervals. The
W intervals calculated using normal quantiles and t quantiles are denoted by W, and Wt
respectively. The results are presented in Tables 4.5 and 4.6. To conserve space in the
tables, we give a group number for each combination of distributions and sample sizes as
identified in Tables 4.3 and 4.4.
w*
(6.91,8.10)
(181,745, 213,023)
wt
(6.89, 8.12)
(181,281, 213,486)
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Group of
1 2 1 Poisson ( Gamma
Distributions
1
Dist of xl Dist of x2
Poisson Log-Normal
3
I 5 I Poisson I Log-Normal
Gamma
4
Gamma
Gamma
I Log-Normal
Gamma
Poisson Log-Normal
Table 4.3: Group numbers for combinations of distributions
Log-Normal
Group of
Sample sizes
Sample sizes of xl Sample sizes of x2
Table 4.4: Group numbers for combinations of sample sizes
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Group
of
Distns.
Group
of Sam.
sizes
Nominal Coverage=90%
Observed Coverage
S wz wt
0.891 0.891 0.900
0.891 0.890 0.899
0.897 0.896 0.903
0.893 0.892 0.902
0.899 0.899 0.906
0.895 0.892 0.902
0.896 0.896 0.904
0.898 0.899 0.906
0.895 0.894 0.903
0.896 0.896 0.904
0.885 0.884 0.893
0.893 0.894 0.901
0.893 0.893 0.902
0.904 0.902 0.910
0.898 0.898 0.905
Nominal Coverage=95%
Observed Coverage
S wz wt
Nominal Coverage=99%
Observed Coverage
S wz wt
Table 4.5: Observed coverage in 10,000 simulations: E(x1)=(1.20, 6.38, 1.74), V(x1)=(1.53,
17.4, 2.36), E(x2)=(1.07, 0.76, 0.79), V(x2)=(0.136, 0.066, 0.130).
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 72
Group
of
Distns.
Group
of Sam.
sizes
Nominal Coverage=90%
Observed Coverage
S wz wt
Nominal Coverage=95%
Observed Coverage
S wz wt
0.936 0.934 0.945
0.946 0.945 0.952
0.944 0.941 0.949
0.948 0.948 0.954
0.946 0.946 0.953
Nominal Coverage=99%
Observed Coverage
wz wt
0.986 0.983 0.988
0.987 0.985 0.989
0.986 0.984 0.988
0.986 0.986 0.989
0.988 0.987 0.992
Table 4.6: Observed coverage in 10,000 simulations: E(xr)=(1.20, 6.38, 1.74),
V(x1)=(1.53,1 7.4, 2.36), E(x2)=(1.07, 0.76, 0.79), V(x2)=( 0.136, 0.066, 0.130).
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 73
Sample size groups 4 and 5 contain the largest sample sizes. From these rows in Tables
4.5 and 4.6, there is evidence that for large samples, all three methods work well. For small
samples (sample size groups 1-3) S and W, coverages are slightly lower than the nominal
value. However, from Wt column, it is apparent that the W intervals calculated using
t quantiles with smallest sample size degrees of freedom, work fairly well even for small
samples.
To compare S and W confidence intervals for normal data, we considered several combi-
nations of variance-to-mean ratios. First, we considered the one product case. The results
are presented in Tables 4.7 and 4.8. In these tables, Adjusted W is denoted by W,.
Means 1 Standard / S:mple 1 Coefl. of Observed Coverage
(Nominal Coverage=.95)
s wz wa
0.948 0.933 0.956
0.949 0.942 0.953
0.949 0.945 0.954
0.950 0.946 0.953
Deviations
Continued on next page
Table 4.7: Observed coverage in 10,000 simulations for normal data. (Nominal
=95%)
Sizes
coverage
Var. of
These simulated results suggest that likelihood ratio intervals are superior to Wald test
intervals for small samples. However, the adjusted Wald test intervals seem to be as good
as likelihood ratio intervals. When the coefficients of variation of the distributions are
substantial, large samples are necessary even for the likelihood ratio intervals to be accurate.
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
Means
P1 P2
Standard
Deviations
61 0 2
Sample
Sizes
Coeff. of
Var. of
Observed Coverage
(Nominal Coverage= .95)
Table 4.8: Observed coverage in 10,000 simulations for normal data. (Nominal coverage
=95%)
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means 75
Results are similar for more than one products. However, then the calculation of likelihood
ratio intervals is much more time consuming. As an example, we generated 5,000 samples,
and computed 95% confidence intervals for sums of three pairwise products of means. The
means and standard deviations for this example were set equal to those in the fish problem.
The results from this simulation are presented in Table 4.9.
I Sample sizes I Nominal Coverage=95% I
Table 4.9: Observed coverage in 5,000 simulations for normal data. (Nominal coverage
=95%)
4.5 Summary
In this chapter we discussed the problem of finding confidence intervals for sums of products
of pairwise means. These confidence intervals were constructed by gathering all the param-
eter values that were not rejected by a formal hypothesis test. We looked at two approaches
in detail; one based on the likelihood ratio test and the other on the Wald test.
In the case at hand, the direct inversion of the likelihood ratio test and the Wald test are
not numerically attractive. Using Madansky's [25] approach, we suggested simplifications to
the computational problem. Related theoretical details were provided for the normal data.
Bartlett- type adjustment factors were computed to improve the accuracy of the coverage
probabilities of the confidence intervals obtained using the Wald test approach. We also
suggested a procedure (Method 2), particularly designed for a product of normal means,
which is numerically more tractable than the approach described under Method 1. While
this approach has the advantage that it can be easily extended to a product of any finite
Chapter 4. Confidence Intervals for a Sum of Pairwise Products of Means
number of means, it has the limitation that it cannot be extended to a sum of products
of means. Suggestions were made for other distributions when the sample sizes are large
enough to justify an appeal to the central theorem.
The applicability of the methods was examined by a Monte Carlo study. For non normal
data, a conservative approach with t quantiles was found to work well even with samples of
moderate sizes. With very large samples, both likelihood ratio and Wald test approaches
seem to work well. In the fish example, the sample sizes were daily aggregates. In order to
produce more precise estimates accounting for daily cycles, it may be necessary to stratify
the day into smaller time intervals. Then, the number of transects within a time interval
may be as small as 30-40. Similarly, when fewer fish are in the river, the numbers from
stationary soundings may be smaller. In this case it may be better t o construct confidence
intervals through W with t quantiles.
For normal data, in situations such as when the coefficients of variation of the distribu-
tions are large, confidence intervals based on Wald test were often found to have smaller
than the nominal coverage. Monte Carlo study shows that the confidence intervals obtained
using the adjusted Wald statistic are as appealing as those obtained from the likelihood
ratio test.
Chapter 5
St rat ified Two-Sample Tag
Recovery Census of Closed
Populations
In the simplest type of tag recovery experiments a simple random sample of m animals is
taken from a population, tagged and released. After allowing enough time for the tagged
animals to mix with others, a second sample is taken and the numbers of tagged animals
( n ) and untagged animals ( v ) are counted. Assuming every animal has the same probability
p of being sampled, and the death and emigration rates are negligible, p is estimated by
p̂ = nlm. The number V of untagged animals in the population at the time of recovery is
then estimated by ? = v/p^ = mvln. This leads to the well known Petersen estimator of
the total population size ? = P + m = m(v + n)/n.
Very often a population is stratified geographically, so that the total population may be
regarded as consisting of separate strata living in different areas. Sometimes, the stratifica-
tion may be with respect to time instead of place. In this case, the population size can be
estimated by a stratified version of the Petersen estimate. In this method, a known number
of tagged animals is released in each stratum, using a different tag for each stratum. After
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 78
allowing for the tagged animals to disperse, a random sample is taken from each recovery
stratum. The numbers of untagged animals and tagged animals of each type in each sample
are then recorded. Several authors have investigated this method. For example, Schaefer
[30] stratified both the tagging and recovery with respect to time to estimate a migrating
salmon run. Chapman and Junge [7] showed that Schaefer's estimator is not in general
consistent. They estimated the numbers of animals in recovery strata from sets of equations
relating to the expected values of observed frequencies. Darroch [13] derived the maximum
likelihood estimators for the case of equal numbers of tagging and recovery strata. For
other cases, he provided alternative estimators. Seber [32](Ch. 1 1), presented an overview
of Darroch's paper. Plante [28] derived maximum likelihood estimators for cases where
the number of tagging strata is not equal to the number of recovery strata. In a similar
experiment, Cormack and Skalski [9] employed log-linear models to analyze coded-wire tag
returns from commercial catches, when the sampling proportions from commercial catches
were known.
In this chapter we closely follow Chapman and Junge, and Darroch, but derive least-
squares estimators of stratum sizes. These estimators are shown to overcome some difficulties
that arise in the existing methods. The estimators are shown to be consistent. As well, the
formulae for calculating asymptotic variances of the estimators are derived.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
5.1 Notation
Let
s = number of tagging strata, and
t = number of recovery strata.
The numbers of animals in tagging and recovery stages are denoted by symbols as in Table
5.1.
Marked
Unmarked
I Total
Tagging stage
Population
Recovery stage
Population I Sample
Table 5.1: Symbols for Numbers of Animals
Subscripts are attached to the above symbols to denote the corresponding tagging or
recovery stratum as follows.
m; = number of tagged animals released in stratum i,
U, = number of untagged animals in stratum i at the time of tagging,
Njj = number of tagged animals migrating from stratum i to stratum j,
Vj = number of untagged animals in stratum j at the time of recovery,
Tj = total number of animals in stratum j at the time of recovery,
n;j = the number of marked animals released in stratum i and recovered in stratum j,
vj = number of untagged animals in j t h recovery sample, and
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
k j = size of sample from stratum j.
The moving, survival, and sampling probabilities are denoted as follows.
8.. 'J = probability that a marked animal moves from stratum i to stratum j , t
4; = x B, = survival probability of a marked animal released in stratum i, j=1
pj = sampling probability in stratum j : assumed non-zero,
pj = l /pj , and
; = Bijpj = probability that a marked animal released in stratum i
will be caught in stratum j.
A sum over any subscript is denoted by replacing the subscript by '+'. For example,
S
m+ = m; = total number of tagged animals released, i=l S
U+ = x U, = total number of untagged animals in tagging strata, i=l
t
N;+ = x Nij = total number of tagged animals successfully migrated to recovery strata j=1
from the ith tagging stratum. t
V+ = x Vj = total number of untagged animals a t the time of recovery, j=1
t
T+ = Tj = total population size, j=1
s t - n;j = total number of tagged animals in the t recovery samples, and n++ -
i=l j=1
t
v+ = vj = total number of untagged animals in t recovery samples. j=1
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
Vectors and matrices are denoted by boldface letters. For example let
Expected values are denoted by an overline. For instance,
- - n = E ( n ) , V = E ( V ) , and V = E ( v ) .
In addition, let
D, = the diagonal matrix formed from the vector, x,
A' = the transpose of matrix A ,
A+ = the Moore-Penrose inverse of matrix A,
1 i f i = j 6 . . =
13 and 0 otherwise,
1 = vector of 1's: the dimension to be understood from the context
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
5.2 Estimation
First, we provide a brief overview of the estimators derived by Schaefer [30], Chapman and
Junge [7], Darroch [13], and Seber [32]. One purpose of this is to point out the difficulties
that can arise in using these estimators. The other purpose is to show the close relationships
between these estimators and the least-squares estimators that we derive in this section.
Schaefer's [30] estimator for the total is
Chapman and Junge [7] showed that this estimator is not consistent in general, but con-
sistent if the samplings in recovery strata are proportional to the population sizes in each
strata. They estimated the unknown stratum sizes (Tj's) from sets of equations relating to
the expected values of the observed frequencies. Schaefer as well as Chapman and Junge
assumed that each animal has zero probability of dying or emigrating between tagging and
recovering, or in other words that the 'survival probability' is one. As this assumption is not
often justified, Darroch framed the likelihood theory in such a way that it may be avoided
if desired. The price paid for dropping this assumption is that one has to be content with
estimating the stratum sizes that would have prevailed had there been no deaths or migra-
tion. These authors discussed the estimation in three cases, according to the relative sizes
of s and t .
Case 1: s = t
If the tagged and untagged animals have the same probability of being captured, then
Observing that
Tj C--E(n; j IN; j ) = Ni+ = rn; for all i, j=1 k j
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
Chapman and Junge formed the set of equations,
which led to the estimator,
Darroch [13] adopted a product-multinomial sampling model to develop the following
probability distributions to derive the maximum likelihood estimators:
and
Under this model, the maximum likelihood estimators of V and qb = O D p are
A
= D v s , and qb = ~ g n
respectively. If p can be estimated, so can V . However, as Darroch pointed out, {Bij)
and {pj) are non-identifiable to the extent of a multiplicative constant, for the likelihood of
{B;j), {pj) is the same as that of {@;j), {pj/y). Hence, neither p nor V can be calculated.
To tie down this non-identifiability, he assumed a common survival probability, 4i r 4 for
all i, and worked with new parameters O* = O/+, p* = 4p, and the corresponding p*, and
V* = V/$. Using the facts that qb = O*Dp*, and 0 * 1 = 1, he derived an estimator for p*
as
Since
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
substitution of (5.6) into (5.7) led to the estimator
A
V* = Dvn-'m.
Consequently,
Darroch pointed out that if the marked and unmarked animals have the same probability of
survival, then (5.9) is roughly the total unmarked size at the time of tagging. When 4 = 1,
the resulting estimator of T+ is the same as Chapman and Junge's total estimator (5.3).
Case 2: s > t
In this case, Chapman and Junge's set of equations (5.2) contain more equations than
the number of unknown Tj's. So, they suggested to estimate the unknown stratum sizes
by pooling enough tagging strata to form a system that has a unique solution. Seber [32]
suggested replacing a set of s equations of t unknown Vj's by a linear combination of them,
giving the estimator,
Here, G is a t x s matrix of rank t such that G n is nonsingular.
Recall that in Case 1, Darroch assumed that 4; z 4 Vi, imposing s - 1 restrictions on
the survival probabilities. When s > t, his approach requires to impose t - 1 restrictions
instead. To see why, note that under assumption of common survival probabilities, there are
s t + t - s independent {qj}, {p;] parameters while there are s t {+;j) parameters. Hence,
when s > t, there are more {$;j) parameters than {qj), {pj*) parameters. Therefore, apart
from the relation + = O*Dp*, there must be other dependencies between the two systems.
Hence, the simple estimation procedure used in Case 1 is no longer valid. So, Darroch took
an alternative approach by allowing 4; to differ. He defined $ = xi 4;/s and worked with
new parameters 4y = I$,/$, 9; = eij/$, pjn = $pj and V* = v/$ . Then, he imposed
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 8 5
t - 1 restrictions on 4;fs. These, including the inherent restriction C, +F"t/ = 1, can be
written in matrix form as
Here, A is a t x s matrix of rank t with each element in the last row equalling l / s , and b
is the 1 x t vector defined as b' = (0, . . . ,0,1). Now, the system {OF), {p?) has the same
number of independent parameters as the system {+;j). Hence, following the same steps as
in Case 1, but using the facts that 1C, = O"Dp*t and AO"1 = b, estimators for pf* and
V" were respectively derived as
- p* = ( ~ D z n ) - ' b , and ?*f = D,(AD:~)-'~.
Case 3: s < t
In this case, Chapman and Junge's system of equations cannot be solved uniquely.
However, they commented that with additional assumptions, the total population size can be
estimated. Darroch also used moment equations to derive estimators in this case. Assuming
that the untagged and tagged animals have the same movement pattern, he formed the
moment equations,
x U i + i j = vj, and
which led to the following set of t equations of s unknown U,'s:
Since there are more than enough equations to estimate U, he suggested replacing the system
of equations by s linear combinations of them or increasing the number of parameters by
relaxing the equality of some of movement probabilities or by introducing some immigration
parameters. Following the first suggestion, Seber proposed the estimator
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
where H is an s x t matrix of rank s such that (Hn') is nonsingular.
In practice, calculating the estimators (5.10), (5.12) and (5.14) might cause problems.
For example, the estimators (5.10) and (5.14) can be influenced strongly by the choice of
matrices G and H. Hence, these estimators can be unappealing unless the matrices G and
H are chosen sensibly. Calculation of (5.12), may be difficult because it requires imposing
restrictions (5.11) on unknown survival probabilities. Since 6;'s cannot be estimated from
our data, extra information is needed to impose these restrictions.
In the next three subsections of this chapter, we closely follow Chapman and Junge, and
Darroch, but derive least-squares estimators. We will show that these estimators are special
cases of the estimators (5.10) and (5.14). They provide sensible choices for matrices G and
H , and thereby avoid the subjectivity that can arise in using estimators (5.10) and (5.14).
They also overcome the difficulties that arise by the requirement of imposing restrictions on
unknown survival probabilities.
5.2.1 Assumptions
To derive the least-squares estimators we make the following assumptions.
1. Animals behave independently of one another in regard to moving between strata.
2. All tagged animals released in a given stratum have the same probability distribution
of movement to recovery strat a.
3. All animals in the j t h recovery stratum behave independently in regard to being
caught, and have the same (non-zero) probability of being caught in the sample.
4. The matrices 5, and n are of full rank. This assumption is required for the consistency
of the estimates. In Section 5.5, we show that this assumption can be relaxed under
certain conditions.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 8 7
5. There are negligible numbers of births, deaths, immigration, or emigration, between
the two sampling times; i.e.,
(In situations like the samplings of dead fish at the spawning grounds, this assumption
means that the numbers of fish that die before they reach the spawning grounds is
negligible.)
This assumption is required in order to estimate the numbers of animals at the recovery
stage. For some animal populations, it may be valid if the time interval between
tagging and recovery are short. But, for many animal populations, this assumption is
not valid. For example, it is certainly not valid if fish are caught in the interim. In
such cases, the investigator has several choices:
(a) Be content with the estimators of scaled numbers of animals at the recovery
stage; e.g. Darroch's estimators (5.8) or (5.12).
(b) Use additional information to estimate the survival probabilities and using these
estimators, unscale Darroch's estimators to get actual numbers at the recovery
stage.
(c) Estimate the numbers at the tagging stage based on Assumption 6 instead.
6. The movement pattern as well as the death and migration rates are the same for
tagged animals and untagged animals in a given stratum.
This assumption is useful in order to derive estimators of the numbers at the tagging
stage. It is more reasonable than Assumption 5 for many animal populations. There-
fore, we will often make this assumption and estimate the numbers of animals at the
tagging st age.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 88
5.2.2 Estimation of the Numbers of Animals in the Recovery Strata (v)
Suppose that Assumptions 1-5 hold. Assumptions 2 and 3 imply that
- E ( v j ) = E I E ( v j l V , ) ] = V j p j , ( j = 1 , ..., t ) , and (5.15)
E ( n ; j ) = m j p j ( = 1 , . . , s and j = 1,2 , . . . , t ) . (5.16)
If an estimator of p were available, then (5.15) could be used to derive an estimator for v. We shall derive estimators for p from (5.16). First note that the system of equations (5.15)
can be written in matrix form as
Dv = DvDp.
The system (5.16) in matrix form is
n = DmODp.
Assumption 5 implies that
Equations (5.17), (5.18), and (5.19) are the three key equations that we use to derive
estimators in this case. Equations (5.18) and (5.19) lead to,
So, it is reasonable to estimate p by a 5 that comes as close as possible to satisfying
n5 = m. (5.21)
When s = t , under Assumption 4 , system (5.21) has a unique solution with high probability.
This solution is given by
5 = n-'m when s = t . (5.22)
When s > t , the system (5.21) does not have a solution since it contains s linear equations
of t unknowns pj. One can at best attain a good approximation. Under Assumption 4, a
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 89
least-squares solution that minimizes the sum of squares of errors,
2 = [(nln)-ln'] m when s 2 t .
Now, note that from (5.17),
By replacing T and p by v and 5 respectively, an estimator of V can be derived as
when s = t
( Dv [(n1n)-'n'] m when s > t.
Consequently, v+ can be estimated by
,. v'n-' m when s = t v+ = 1'V =
V' [(n1n)-'n'] m when s > t.
When s < t, p cannot be estimated from the system (5.21). Therefore, V also cannot be
estimated. However, as Chapman and Junge [7], and Darroch [13] pointed out, an estimator
of U is available if Assumption 6 is satisfied. This estimator is derived in the next subsection.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 90
5.2.3 Estimation of the Numbers of Animals in the Tagging Strata (U)
Suppose that Assumptions 1-4 and 6 hold. Under Assumption 6,
E(vj) = u,fIijpj ( j = 1, . . . , t), i
or in matrix notation,
This is the first key equation that we use in this case to derive estimators. The second key
equation is (5.18). That is,
Now, obtaining
qb = O D - D-'5 P - m
from (5.28), substituting in (5.27), and taking the transpose, yields
- 4 v = n D ~ U .
Now, let
We propose to estimate U by 6 that minimizes the sum of squares of the errors,
Under Assumption 4, this estimator is given by
D,[(nnf)-'n]v when s < t 6 = ( ~ ~ ( n ' ) - ' v when s = t
The resulting estimators of the total untagged animals are,
A - v'[nl(nn')-']m when s < t u+ = U ' l = vl(n)-' m when s = t .
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
5.2.4 Some Remarks
Remark 1
Suppose 01 # 1 , but 4; = 4 V i. Then, following Darroch, we can define scaled parameters
0:' = 0,/4, p; = 4pj, and = vj/$. Then, the new parameters satisfy key equations
(5.17), (5.18) and (5.19). Therefore, (5.25) now estimates v. When s = t, estimator (5.25)
is essentially the same as Darroch7s estimator (5.8). However, this is not particularly useful
unless the survival probabilities can be estimated.
Remark 2
The estimators (5.25) and (5.32) are special cases of (5.10) and (5.14) respectively. These
suggest that G = n1 and H = n are sensible choices to be used in (5.10) and (5.14)
respectively.
Remark 3
According to Searle [31], the Moore-Penrose inverse nt of n is a unique matrix such that
(i) nn tn = n,
(iii) nn t is symmetric, and
(iv) n t n is symmetric.
One can show (e.g., Albert [I], pp. 20-21) by direct verification of (i)-(iv) that if n is of
full rank, then
n'(nnt)-I when s < t
when s = t
(nln)-In' when s > t .
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
Therefore, the estimators derived above can be summarized as follows.
Remark 4
In deriving the least squares estimates 9 and 6, we have neglected the variance-covariance
structures of c;'s and C3s. Better estimates may be derived by taking these into account.
\
= ntm when s 2 t, A - V = ~ , n ~ m when s 2 t, A - V + = v'ntm when s 2 t,
6 = ~ , ( n ' ) ~ v w h e n s < t , a n d
o+ = v'ntm when s < t. /
In order to calculate the variance-covariance matrix of E , we should adopt a suitable
probability model. Under Assumptions 1-3, probability di~t~ributions (5.4) and (5.5) provide
such a model. For this model, the variance of c; = C&l n;jpj - m, is
+
The covariance
estimators may
terms are zero; i.e, Cov(e;,q) = 0 Qi # 1. Weighted least-squares (WLS)
be derived by minimizing
where
with 4ij's defined by
These estimates can be obtained by solving the estimating equations,
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 93
Since the weights are functions of unknown pj's, these estimating equations are biased.
However, an estimate which is unbiased conditional on n , can be derived using an iteratively
reweighted least-squares method.
Let C ( p ) be the diagonal matrix with diagonal elements 6:'s. Suppose that po is an
initial guess for p. So, a t this point, the initial guess of the variance-covariance matrix is
given by C ( p o ) . Let
z0 = m - npo, and
So = [m - np]' X-'(po) [m - n p l -
*
Now, we minimize So with respect to p. The requirement, = 0, leads to
n'C-'(p0)n5 = n ' z - ' ( p o ) m (5.39)
= n ' ~ - ' ( p o ) [zo + W o l ,
n'z- ' (po)n [? - pol = n ' X - ' ( p o ) ~ O , and hence
6 = po + [nlC-'(po)n] -' nlX-'(&)zo.
This gives rise to the iterative equation,
with zk defined as
This procedure typically does not minimize S . But it converges, when it converges, t o a
root of the equation (5.39) with po replaced by p; i.e.,
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 94
This estimating equation is unbiased conditional on n under the model which justifies (5.36).
In order to derive a similar estimate for U, we need the variance-covariance structure
of C. Recall that the estimator 6 was derived under Assumptions 1-4 and 6. Under these
assumptions, it is reasonable to assume a probability model for the unmarked animals that
is parallel to (5.4). I.e,
where
v,j = the number of unmarked animals which are in the ith stratum at the tagging stage
and in the j t h recovery sample.
Then, from (5.31)
Therefore, under probability models (5.5) and (5.42),
Let C(U) be the matrix (yij), but with the $;j's replaced by estimators given by (5.38).
Now, suppose that Uo is a suitable initial guess for U. Then, by similar arguments to those
above, the iterative procedure,
can be used to derive an estimate for U.
In the subsequent sections, we refer to the estimators derived via iterative procedures
(5.41) and (5.43) as iteratively reweighted least squares estimates (IRLSE).
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 95
R e m a r k 5
When s > t, by premultiplying (5.20) by ii' , and performing simple matrix manipulation
under Assumption 4, p can be expressed as
p = [(ii'ii)-lii'lm. (5.44)
Substitution of (5.44) in (5.24) yields
- V = ~ ~ [ ( i i ' i i ) - l i i ' ] m when s > t . (5.45)
Consequently,
- V + = 1'9= $[(i?ii)-'-ir']m when s > t .
Similarly, when s 5 t, by premultiplying (5.30) by ii, and performing simple matrix
manipulation under Assumption 4, U can be written as
Hence,
Now, note that under Assumption 4,
--I ---, -1 n ( n n ) w h e n s < t
when s = t
i i ' ) w h e n s > t .
Therefore, the above quantities can be written as \
p = d m when s > t , - V = D$m when s > t, - V + = g i i t m when s 2 t,
U = ~ , ( i i ' ) t i f when s < t, and
U+ = giitm when s 5 t. /
+ (5.50)
These representations are useful in proving the consistency of the estimators in Section 5.3,
and calculating the variances in Section 5.4.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
5.3 Consistency
As Chapman and Junge pointed out, the consistency of estimators based on samples from
finite populations has been variously defined. According to one such usage, an estimator A
X of parameter X would be called consistent if = X whenever the sample taken without
replacement exhausts the population. However, from a practical point of view, in a study of
populations that number several hundred thousands or millions (e.g. salmon populations),
it is unreasonable to think of a sample equaling or nearly equaling the population size. Yet,
at the same time, it is possible that the samples are random and very large so that the
weak law of large numbers should be applicable. Hence we consider the limit process: { O j j ) ,
{pj} constant, all m; t oo and V, -t oo such that m;/m+, V,/V+ and m+/V+ approach A
fixed constants. We say that V is a consistent estimator of if under these conditions, A - - Vj/Vj -. 1 in probability for all j. Similarly, we say that 6 is consistent for U if @;/u; + 1
in probability for all i.
First notice that under probability mode1 (5.4), and in the above limit process, n;j/fi;j +
1 in probability, for all i and j. In other words, n is consistent for ii. The matrix product
is a continuous function and the matrix inverse is locally continuous in a neighborhood
of an invertible matrix. Therefore, under Assumption 4, the Slutsky-Frechet theorem (see
Appendix) implies that the Moore-Penrose inverse nt of n is consistent for the Moore-
Penrose inverse fit of ii. This, in turn implies that 5 = n t m is consistent for d m , which
by (5.50) is equal to p.
Next, notice that under probability model (5.5) and the above limit process, v is con-
sistent for 7. Further notice that the only random variable involved in 5 is n. Since v and
n are independent by Assumptions 1 and 3, so are v and 6. Therefore, = ~ , n t m is
consistent for ~ & m , which by (5.50) is equal to V. A similar argument will show that 6 is consistent for U.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 9 7
5.4 Variance
Darroch used probability distributions (5.4) and (5.5), to derive the formulae for the ap-
proximate variances of his estimators. Accordingly, when s = t ,
A
Var(p*) = C x X-'D,D~(X-')~, (5.51)
v a r ( V ) x DvXDy + DV(Dp - I), and (5.52)
4 v a r ( Q ) x v CF + V1(p - I). (5.53)
Here, p; = Cj Oijpj - 1.
In the next two subsections, we derive formulae for the variances of estimators (5.35).
A
5.4.1 Variance of
Recall from (5.50), (5.34) and (5.35) that when s > t , - V = Dvp, and A
V = D ~ < = ~~n~ rn = Dv [(nln)-'nl] rn.
Now let
Then,
Y = (V - F).
A - V - V = D d s - p ) + D y s .
The only random variable involved in 5 is n. Since v and n are independent by Assumptions A
1 and 3, so are y and 5. Furthermore, E(y) = 0. Therefore, the bias of V is
A
E(V - V ) = DvE(s - p). (5.57)
A
The mean squared error of v is
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 98
Again, because y and 5 are independent and E ( y ) = 0, the last term is zero. Hence,
E - ) ( - ) I ] = DvE [ ( 5 - p)(* - p)'] DT + E [Dy5$Dy] . (5.58)
A
We need the bias and the mean squared error of 5 to calculate those of v. So, let us
calculate those quantities first.
Calculation of bias and mean squared error of 5
This calculation contains the following steps.
Step 1. Recall that when s 2 t ,
Step 2. Let
Show that
5 = p - iitxp + a random quantity with expected value ~ ( r n ~ ' ) . (5.61)
Thereby show that the bias of 5 is O(rn7'). The expression (5.61) is obtained by
expanding the inverse matrix in (5.59) using the following identity:
If A = A + Z and both A and A are nonsingular, then,
A-' = [I - A-l~1A-l + ( A - ~ z ) ~ A - ~ . (5.62)
Step 3. Show that the leading term in the approximate mean squared error is
which is O(rn~') . Hence, the bias is asymptotically negligible compared to the root
mean squared error, which is ~ ( r n ; " ~ ) , and (5.63) can be considered as the approxi-
mate variance-covariance matrix of 5.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 99
Now, we present the detailed calculation in Step 2. Let X be defined as in (5.60). Then,
n'n = (ii' + X1)(ii+ X)
4- = nn+ii 'X+X'i i+XX'X
= r + z ,
where
Hence, substitution of'A = (n'n) and A = I' in (5.62) yields
Therefore,
nt = (nln)-'nl = [I - r-'Z]I'-'(Ti' + X') + ( I ' - l ~ ) ~ n ~ . (5.67)
Post-multiplying (5.67) by m and substituting ii'X+X1ii+X'X for Z in the square bracket,
leads to
where
b = I'-'(ii'x + ~ ' i i ) I ' - ' ~ ' r n + r- ' (~'~)I ' - l ( i i ' + X1)m, (5.70)
and
Since E(X) = 0, (5.69) implies that the bias, E(6) - p = E(-b + r). Now, we show
that this bias is ~ ( r n ~ ' ) . For this, consider the limit process: {Oij), { p j ) constant and
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 100
all m; + oo in such a way that m;/m+ approach fixed constants. First, note that ii =
O(m+), I'-' = ~ ( r n ~ ~ ) in this limit process, and m = O(m+).
Next note that since (n;l, n;2, . . . , nit) are multinomial random variables, E(XijXlk) =
6,10(m+) and E(XijXlkXab) = 6,&O(m+). Hence,
Now, in order to find out the order of E(r) , let
Then,
Using Jensen's inequality and Cauchy-Schwarz inequality (see Appendix), we can see that
In F,j, the contribution from I?-' is ~(m;') . So, its contribution to E(F$) is ~ ( m ; ~ ) . The
terms in Z that contribute the largest order terms to E(Ffk) are K'X + X'Z. So, the highest
order of moments of X generated in <; by these terms is 4. Since these are multinomial
moments, the maximum contribution is O(m:). Finally, the highest contribution of terms
ii that come from Z is O(m:). Therefore, the terms E[Ffk] are O ( ~ T ~ + ~ + ' ) = 0 ( m+ -2).
Furthermore, the order of E(6:) is the same as that of $, which is O(1). Therefore,
E ( r ) = [ ~ ( r n ; ~ ) ] ' I 2 = ~(rn;'). (5.73)
Order expressions, (5.72) and (5.73) imply that the bias of 5 is ~(m; ' ) .
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 101
Now, we proceed to Step 3. From (5.69), the leading term in E [(g - p)(z - p)')] is
Note that lit = 0 ( m ~ ' ) , second order moments of X are O(m+) and p = O(1). Hence, C
is ~(m; ' ) , and the bias of 5, which is ~(m; ' ) , is negligible compared to the root mean
squared error. Therefore (5.74) is the approximate variance of 5. A
Calculation of the mean squared error of V A
First, in order to find the approximate bias of V, augment the above limit process by
supposing that all Vj -, oo in such a way that Vj/V+ and m+/V+ approach fixed constants. A
Equation (5.57) gives the bias of V as
A
E(V - V ) = DvE[5 - p]. (5.75)
A
Since i 7 is O(V+) and the bias of 5 is ~(m; ' ) , the bias of V is O(V+/m+) = O(1) in the
above limit process.
A
As given by (5.58), the mean squared error of is
The first term in (5.76) is DvCDV, where E is as given by (5.74). This is O(V:)O(~;') =
O(V+) = O(m+) in the augmented limit process.
Substituting p - l i t x p - b + r for 5, the second term can be written as
E [DYI3I3'~,] = E [DypplDy] + 0(vc1) .
A
This leading term is O(V+) = O(m+) in the limit process. So, the mean squared error of V A
is O(m+). Since the bias of V is 0(1), it is negligible compared to the root mean squared A
error. So, the approximate variance of V is
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 102
Estimation of variances
First, to calculate (5.74), note that the (i, 1)th element of EIXpplX'] is Cjk pj~kEIXijXlk].
According to the probability models (5.4) and (5.5),
Therefore,
where pi = Cj Bijpj - 1 = C. 3 +ijp! - 1. Noting that this is the (i, 1)th element of DmDp,
the approximate variance (5.74) of 6 can be written as
Next, in order to calculate the second term in (5.77) note that
Hence,
E(DypplDy) = DV(Dp - I). (5.82)
Then, (5.80) and (5.82) lead (5.77) to
var(?) r~ DyCDv + Dy(Dp - I), and (5.83)
var(P+) r~ V'XV + V1(p - 1). (5.84)
The approximate variances may be estimated by replacing the unknown parameters by
their estimates. An estimator for p is,
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
Remark 6
When s = t, Darroch's formula (5.51) can be deduced from (5.80) by noting that iit = ii-'.
Consequently, (5.52) and (5.53) are also equivalent to (5.83) and (5.84) respectively.
5.4.2 Variance of 6
Now, we derive a formula for the approximate variance of 6 given by (5.35). Since the
derivation is similar to that in Subsection 5.4.1, here we present only the main steps.
When s 5 t, let
B = Dm(iii?)-'ii = Dm(*)', and
B = Dm(n n')-'n = ~ ~ ( n ' ) ~ .
Then, from (5.50) and (5.35),
U = BJ, and
6 = 6 v
respectively. Let y be defined as in (5.55). Then,
6 - u = (B - B)v+ By, and
var( f i ) r~ E [(B - ~ ) i i i i ' ( B - B)'] + E [BYY'B']
= E [(B -B)V+(B - B)'] + B E [yyl] B'. (5.90)
Letting v a r ( 6 ) = n, E [(B - B)BV'(B - B)'] = A, and E [yy l = Y, respectively, we
have
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 104
Calculation of A
Let X be defined as in (5.60). Then,
nn' = (E+ X)(ii'+ X')
= ETi'+Ex'+Xi?++X1
= r + z ,
where I' = iiTi' and Z = EX'+ Xi? + XX' (Note the new definition for I'. It was previously
defined as I' = Z' ii). Then, using the identity (5.62),
Pre-multiplying (5.92) by Dm and substituting EX' + XX' + XX' for Z in the square
bracket,
A
B = D ~ ~ - ~ H + D ~ I ' - ' x ( I - H ~ E ) - D ~ I ' - ' Z X ' ~ - ' Z + R
= B + AXQ - ~ ~ ' ( i i ' ) ~ + R, (5.93)
where
and R is the remainder term with expected value of order ~ ( r n ; ' ) . Post-multiplication of
(5.93) by V leads to
(B - B)V m AXQV - BX'(X')~J, and
A = E [(B - B )BV'(B - B )'I i~ A E [XQBJ'Q'X'] A' + BE [xt(Zi') 'V iif(ii) 'XI B'
-A E [xQiW'iitx] B' - BE [x'(T~')~~W'Q'X'] A'
= A E [ X a a ' ~ ' ] A' + BE [X'PP'X] B'
-AE [XaP'X] B' - BE [X'Pa'X'] A', (5.94)
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 105
where
a = QV, and
p = (Yip .
Now letting
D = E [Xaa'X'] , F = E [X'PP'X] , and
G = E [XaP'X] ,
and noting that Q = 0 when s = t, A can be written as
ADA' + BFB' - AGB' - BG'A', if s < t, and
BFB', if s = t.
Calculation of ~ a r ( 6 + )
From (5.35),
But, since this is a scalar, it can also be written as
6+ = mt(n')tv.
Similarly, from (5.50),
Letting
b' = mt(ii')t , and
b = m'(n1)t,
(5.96) and (5.97) can be written as
6+ = b'v, and
U+ = b ' ~ .
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 106
Comparing (5.89) and (5.88) with (5.100) and (5.101) respectively, we can see that the
calculations in Subsection 5.4.2 can be repeated with U and B replaced by U+ and b'
respectively, to calculate the variance of 6+. This also requires replacing A by
Consequently, a formulae for the variance of 6+ can be obtained by replacing A and B in
the variance formula for 6 by a' and b' respectively. Letting
a'Da + b'Fb - a'Gb - b'G1a, if s < t, and
b'Fb, if s = t,
the approximate variance of 6+ can be written as
Est imation of variances
Since E ( X i j X l k ) = 6irmi[6jk$;j - $ij$ik], the matrices D , F and G can be evaluated as
follows:
and
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations
Now we evaluate
Since p cannot be estimated when s < t, the expression (5.81) cannot be used to estimate
Y. Instead, we should express it in terms of the estimable quantities, $;j. According to
(5.81), Y is a diagonal matrix. Under probability models (5.5) and (5.42), the j t h diagonal
element of Y is
= Var C v;j l;:, I
Now, the approximate variances may be estimated by replacing all the parameters by their
estimates.
5.5 Relaxing Assumption 4
So far, all the calculations were done under the assumption that Ti and n were of full rank
(Assumption 4). As long as Assumptions 1-3 and 5 or 6 are satisfied, this is sufficient for
the estimators to be consistent. This assumption was also useful in deriving formulae for
the approximate variances of the estimators.
According to Seber [33](pg. 327), and Albert [l](see Appendix), regardless of whether
or not X is of full rank, f i = xty minimizes Ily - XPII. So, regardless of whether or not
n is of full rank, (5.35) provides least-squares estimates. But now, nt is not necessarily
defined by (5.34). In general, nt can be calculated via a singular-value decomposition. Even
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 108
though least-squares estimates are available in this way under any circumstances, they may
not necessarily be consistent.
Recall from (5.18) that
The matrix ii has full rank if and only if O is of full rank. If ii has full rank and the samples
are large, a non-full rank matrix n is extremely unlikely. So, as long as O has full rank, and
samples are large, we can estimate the numbers of animals in each of the strata (tagging
or recovery, depending on relative sizes of s and t, and on validity of Assumptions 5 or 6)
consistently.
As Darroch noted, there are a few obvious instances where the O matrix is not of full
rank. For example,
1. Ojj = 0 Vi,
2. Oij/Olj is constant independent of j , and
3. Oij/Ojk is constant independent of i.
In the first instance, the j t h stratum is effectively non-existent, and can be ignored. In
the other two instances, the relevant strata may be pooled to avoid the non-full rank ii
matrices. In Chapter 6, we show that such poolings do not affect the consistency of the
pooled estimators. So, even if O is of less than full rank, we are still able to consistently
estimate numbers of animals in unpooled strata and totals in pooled strata. It is possible
to test for the above two collinearities in the O matrix as follows.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 109
Test for collinearity of two rows of O matrix
Tagging
stratum
# marked
released
Total
nil ni 2 . . . nit I m, - n;+ I
# marked recovered in stratum j
1 2 . . . t
m;+ml nil + nil ni2 + nl2 . . . nit + nrt 1 mi + ml - (ni+ + nl+) I
# marked
not recovered
We are interested in testing the hypothesis Ho : O i j / O l j =constant for all j . Equivalently,
we can test
Under Ho, for a = i or I ,
Let e;j be the expected number of marked animals which are released in stratum i and
recovered in stratum j. Then,
Under Ho, the test statistic
has an approximate X 2 distribution with t - 1 degrees of freedom.
As Darroch pointed out, the goodness of fit of the special hypothesis Ho : 9ij = B l j V j
is equivalent to a test of homogeneity for the rows of 2 x ( t + 1 ) contingency table including
the last column in the above table. Now, under the null hypothesis, T, has an approximate
X 2 distribution with t degrees of freedom.
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 110
Test for collinearity of two columns of O matrix
Similarly, the hypothesis Ho : O;j/O;k=constant for a l l i, can be tested using the test
statistic
Here,
n+,(n;j + nik) e;, =
n+j 4- n+k
Under Ho, T, has an approximate X2 distribution with s - 1 degrees of freedom.
5.6 Discussion
In this chapter we addressed the problem of estimating the population size using stratified
two-sample tag recovery data. We derived least-squares estimators and provided algorithms
for obtaining weighted least-squares estimates. These estimators have the advantage that
they can be easily applied even when the numbers of tagging strata and recovery strata are
unequal.
However, these estimators share some weaknesses that are inherent to the existing
methodology. For example, the unrestricted nature of estimation procedure can lead to
probability estimators that are out of range, and population size estimators that are nega-
tive. These problems are particularly likely to arise when the data are sparse or the model is
inadequate. Although the deficiencies should be dealt with directly, it is possible to impose
direct restrictions on estimates of population sizes and probabilities. For instance, the pro-
cedure for estimating neglected that 0 < pj < 1 tl j . This restriction can be incorporated
into the estimation procedure by minimizing the error sums of squares subject to constraints
or by converting the problem into an unconstrained one. The latter may be accomplished by
re-parameterizing the problem using aj = log (pj/ l - pj) and minimizing the sum of squares
of the errors, E; = CiZ1 nij(l + e-"1) - m;, with respect to the aj's. Similarly, in estimating
Chapter 5. Stratified Two-Sample Tag Recovery Census of Closed Populations 11 1
U, we neglected that U; > 0 V i. This restriction may be imposed by writing U, = eQi and
minimizing the sum of squares of errors with respect to these a;'s. This approach does not
necessarily provide reasonable estimates. However, it avoids unacceptable estimates.
Example 1 in Section 6.6 shows that least-squares estimation leads to reasonable esti-
mates which are close to those obtained by other methods with more assumptions.
Chapter 6
Pooling in a Stratified
Two-Sample Tag Recovery Census
Investigators often pool strata for various reasons. The pooling can occur before or after the
experiment. For instance, if the investigators do not know how the population is stratified,
then they may conduct an unstratified experiment and use the Petersen estimator. Also, if
they are unable to use different tags for some of the tagging strata, then those strata may
be pooled together. Investigators often pool data after the experiment when the numbers
of tagged animals recovered in some strata are small. One such pooling can be found in
Darroch's [13] paper, where he used Schaefer's sockeye salmon data set to estimate a run of
migrating salmon. This data set has eight tagging strata and nine recovery strata. Darroch
pooled the first three and last three weeks of tagging into single strata and the first three
and last four weeks of recovery into single strata because frequencies in these weeks were
too small to be used in large sample theory. As pointed out in Section 5.5, experimenters
may also pool strata to avoid non-full rank matrices.
In any case, when the strata are pooled, the estimators can be inconsistent. For example,
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
the Petersen estimator,
is not in general consistent. Darroch showed that (6.1) is consistent under each of the
following practically appealing conditions:
i. A constant proportion of each recovery stratum is sampled (pj = p for j = 1, . . . , t).
ii. A constant proportion of each stratum is tagged (m;/V, = m+/V+ for i = 1, . . . , s)
and the movement pattern for tagged animals and untagged animals is the same.
iii. Complete mixing of the whole population ( O j j = O j for all i and j) and the movement
pattern for tagged animals and untagged animals is the same.
In the next three sections, we extend Darroch's work by finding sufficient conditions
for pooling strata partially, so that the estimates may be consistent. We first consider
pooling tagging and recovery strata separately, and finally combine the results to facilitate
both types of pooling simultaneously. Then in Section 6.5, we examine the biases that can
occur in the estimators if these conditions are not satisfied. Throughout the rest of this
chapter, the pooled quantities are denoted by attaching a superscript '*'. The numbers of
tagging and recovery strata after pooling are denoted by s* and t* respectively ( '*' used
in the subsequent sections should not be confused with the '*' used previously to describe
Darroch's estimates.)
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
6.1 Partial Pooling of Tagging Strata
Suppose that two of the tagging strata are pooled. To simplify the notation, without loss
of generality, renumber the strata such that the pooled strata are the last two. The pooled
quantities are,
Naturally, one would use
A - V = ~ , , ( n * ) ~ r n * when s* > t .
and
C* = Dm*([n*It)tv when S* 5 t , (6.3)
to estimate v and U* respectively. Here, n*+ is the Moore-Penrose inverse of n* defined
similarly to (5.34).
Arguing as in Section 5.3, it can be shown that (6.2) and (6.3) are consistent for
Ddfi;*)trn* and ~,*([iT+l')tV respectively. So to derive sufficient conditions for the consis-
tency of (6.2) and (6.3), we find conditions under which D&P)trn* = v and D , * ( Z ) ~ =
U* respectively. For this, it is enough to find conditions under which the pooled quantities
satisfy the key equations in Sections 5.2.2 and 5.2.3. Then, by the same line of calculations
as in these Sections, and in Remark 5, the consistency of the pooled estimators will follow.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 115
Result 6.1 Suppose that two of the tagging stmta are pooled. Also suppose that Assump-
tions 1-3 are satisfied and matrices i? and n* are of full rank.
R.6.1.1
If Assumption 5 is satisfied, and s* 2 t , then the estimator given by (6.2) is consistent
for V.
R.6.1.2
If Assumption 6 is satisfied, and s* 5 t , then the following Conditions 1 and 2 are
individually suficient for the estimator given by (6.3) to be consistent for U*.
Condition 1
The ratios of the numbers of tagged to untagged animals are the same in the two pooled
stmta. Le,
U; = cm; for the two pooled tagging strata.
(This is the analogue of Darroch's suficient condition (ii) for the Petersen estimator
to be consistent.)
Condition 2
The patterns of moving from pooled tagging stmta to any given recovery stratum are
the same. Le.,
B(s- l ) j = Bsj for all recovery stmta j .
(This is analogous to Darroch's condition (iii)).
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Proof
To prove R.6.1.1, it is enough to show that the pooled data satisfy the three key equations
in Subsection 5.2.2. Since we pool only the tagging strata, the first key equation (5.17) in
Subsection 5.2.2 is unchanged. That is,
Next, we show that there exists a matrix 6 , such that
rr* = D, .~D, ,
and
For this, let
where &s-l)j's are probabilities such that x;=, 5 1. Then, for i = 1,. . . , s - 2 and
j = 1, . . . , t ,
and for j = 1,. . . , t ,
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Therefore, if
then, 6 satisfies (6.5). Further note that if O satisfies (5.15), then so does 6; i.e., it
satisfies (6.6). Therefore, from (6.4), (6.5), and (6.6), by the same line of calculations as in
Subsection 5.2.2 and Remark 5, it follows that = ~ d i P ) t m * . This proves R.6.1.1.
To prove R.6.1.2, we find conditions under which the pooled data satisfy the two key
equations (5.27), and (5.28) in Subsection 5.2.3. Note that for j = 1, . . . , t,
s-2 ~ ~ - ~ e ( ~ - , , , + u e = C ~ i ' e i j p j + ( U S - I + U S ) ' 'J ]
i=l [ U s - l + u s
This can be written as
U.,- le(s- l) j t uses j - I = O ( s - l ) j for j = 1, . . . , t .
We have already shown that the second key equation (5.28) in Subsection 5.2.3, which is
identical to (5.18) is satisfied by the pooled data if (6.8) is satisfied. So, both key equations
in Subsection 5.2.3 are satisfied by the pooled data if
I for j = 1, . . .,t. (6.12)
Conditions 1 and 2 individually satisfy (6.12). This completes the proof.
R e m a r k
As pointed out in Section 5.5, investigators may have to pool some tagging strata if the
O is non-full rank due to collinearity of the rows. As we showed in Result 6.1, the pooled
estimates are consistent if the corresponding rows in the O matrix are equal. However, if the
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 118
rows are not equal, but collinear, then such pooling does not necessarily lead to consistent
estimates of U (although the estimate of is still consistent). Recall that 6' is consistent
if (6.12) is satisfied. From this we can see that the estimates 6' obtained by pooling tagging
strata corresponding to two collinear rows of the O matrix are consistent if Condition 1 is
also satisfied. That is, if
e i j / e l j = a constant not equal to 1, and
Ui = cmi,
for some tagging strata, then the investigator can pool those tagging strata to get consistent
estimates of U.
6.2 Partial Pooling of Recovery Strata
Now, we consider the partial pooling of recovery strata. Suppose that two of the recovery
strata are pooled. Without loss of generality, renumber the recovery strata such that pooled
strata are the last two strata. Then, the pooled quantities are
and
Naturally, one would use the estimators,
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
A*
V = ~ " . ( n * ) ~ r n when s > t* ,
and
6 = ~ ~ ( n * ' ) ~ v * when s 5 t*,
to estimate V and U respectively.
Result 6.2 Suppose that two of the recovery strnta are pooled. Also suppose that Assump-
tions 1-3 are satisfied and matrices i? and n* are of full rank.
(a) If Assumption 5 is satisfied, and s 2 t*, then the following Condition 3 is SUB- cient for the estimator given by (6.1 6) to be consistent for v.
(b) If Assumptions 5 and 6 are satisfied, and s > t*, then the following Condition 4
is suflcient for the estimator given by (6.16) to be consistent for v.
R.6.2.2
If Assumption 6 is satisfied, and s 5 t*, then the following Conditions 3 and 4 are
individually suficient for the estimator given by (6.17) to be consistent for U .
Condit ion 3
The sampling proportion is the same for both pooled strnta; i.e, pt-1 = pt. (This
is the analogue of Darmh's suficient condition (i) for the Petersen estimator to be
consistent.)
Condit ion 4
The probabilities of moving to pooled recovery strata from any given tagging stratum
are constant multiples of each other; i.e., 8;t/8;(t-1) = k (a constant) for i = 1, . . . , s.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Proof
Let
for some 0 < 5 1.
Note that if
then
Next, note that for i = 1, . . . , s and j = 1 , . . . ( t - 2 )
E(nTj) = E(nij) = miOijpj = miBrjpj,
and for i = 1,. . . , s
E n 1 ] = E(n+l) t nit) = m;&(t-l)~t-l t mi&tpt
Hence, if
then
8i(t-l)pt-l + Bitpt i - 1 = [ I for i = 1,. . . , s ,
oi(t-1) + 8it
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
I.e, if - vt-1pt-1 - + V t p t + &tpt I , for i = 1, ..., s
V t - 1 + V t ei(t-1) + 8it
then pooled data satisfy key equations in Subsection 5.2.2. Since Condition 3 satisfies (6.25),
this condition is sufficient for the the pooled data to satisfy the first two key equations in
Subsection 5.2.2. If Assumption 5 is satisfied, the third key equation is satisfied by a*. This proves part (a) of R.6.2.1.
To prove part (b) of R.6.2.1, suppose that both Assumptions 5 and 6 are satisfied. Then
f o r j = 1, ..., t - 2 ,
and
If (6.23) is satisfied, then (6.27) and (6.28) can be written as (6.20). We have already
shown that (6.24) is satisfied if (6.23) is satisfied. So, now, it is sufficient that
q t - 1 ) ~ t - 1 + &tpt 1 = a constant for i = 1,. . . , s , ei(t-1) + Bit
for the pooled data to satisfy the key equations in Subsection 5.2.2. Condition 4 satisfies
(6.29). This proves part (b) of R.6.2.1.
Next, to prove R.6.2.2, recall that (6.24) is valid if (6.23) is satisfied. Also, note from
(6.26) and (6.28) that if (6.23) is satisfied, then
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 122
In other words, both two key equations in Subsection 5.2.3 are satisfied if (6.29) is satisfied.
Conditions 3 and 4 both individually satisfy (6.29). This proves R.6.2.2.
6.3 Partial Pooling of Tagging and Recovery Strata
Usually, in practice, the investigator may need to pool tagging strata as well as recovery
strata. Then U * and V- will be estimated by
1 8
V = ~ , p ( n * ) ~ m * when s* > t*, (6.31)
and
6* = ~ ~ * ( n * ) ~ v * when s* 5 t*. (6.32)
to estimate and U respectively. We now combine Results 1 and 2 to derive sufficient
conditions for these estimators to be consistent.
Result 6.3 Suppose that two of the tagging strata and two of the recovery are pooled. Also
suppose that Assumptions 1-3 are satisfied and matrices P and n* are of full rank.
(a) If Assumption 5 is satisfied, and s* 2 t*, then Condition 3 is sufficient for the
estimator given by (6.31) to be consistent for V-.
(b) If Assumptions 5 and 6 are satisfied, s* 2 t*, then Condition 4 is sufficient for
the estimator given by (6.31) to be consistent for V'.
R.6.3.2
If Assumption 6 is satisfied, and s* 5 t*, then the following combinations of conditions
are sufficient for the estimator given by (6.32) to be consistent for U * .
(a) Conditions 1 and 3.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
(b) Conditions 1 and 4.
(c) Conditions 2 and 3.
(d) Conditions 2 and 4.
So far we have considered pooling two tagging strata and two recovery strata only. But,
investigators often pool more than two tagging and recovery strata to form more than one
group of pooled strata of each type. Sufficient conditions for such poolings can be derived
from Results 6.1, 6.2 and 6.3. In such cases, the conditions should be true for each group
of strata that is pooled.
We can use Results 6.1, 6.2 and 6.3 to verify Darroch's conditions (for the particular
case where all the tagging and recovery strata are pooled). To see this, first suppose that
Assumptions 1-3 and 5 are satisfied and Darroch's condition (i) is true. Then, R.6.3.l(a)
implies that c+ is consistent for V+. Secondly suppose that Assumptions 1-3 and 6 are
satisfied and Darroch's condition (ii) or (iii) is true. Then, by R.6.1.2, all the tagging strata
can be pooled without affecting consistency of 0:. At this point, the corresponding 6 is 1 x t , and automatically satisfies Condition 4. So, by R.6.2.2, all the recovery strata can
be pooled to get a consistent estimator of U+. If in addition Assumption 1 is satisfied, this
estimates V+.
6.4 Variances of Pooled Estimates
Having listed various conditions under which the pooled estimates are valid, let us now
suppose that the investigator correctly assumes that one of them is satisfied and conducts
an appropriate pooling. Although the investigator can consistently estimate the number of
animals by substituting the pooled data in an estimator derived in Section 5.2, the variance
formulae derived in Section 5.4 may no longer be valid for the pooled data. We now examine
sufficient conditions for those formulae to correctly estimate the variances. The following
discussion considers the variance formula for 9. The complicated form of the variance
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 124
formula for 6 causes difficulties in determining the effect of pooling on that formula, and
is not considered here.
Complete Pooling
First, suppose that one of Darroch7s conditions is valid, and an unstratified experiment has
been conducted. According to Darroch, then the approximate mean squared error of the
Petersen estimate is
where v = E(v+) = C . 3 V . )PI . and 7 = E(n+) = Cij mifl;,jpj. In order to estimate the mean
squared error, the investigator has no course but to substitute estimates:
in (6.33). Darroch showed that this leads to overestimation, unless the sampling probabilities
(the pj7s) are equal.
Partial Pooling
Now suppose that some of the tagging and recovery strata are partially pooled under suffi-
cient conditions provided by Result 6.3. Then, the calculations (5.54)-(5.77) in Subsection
5.4.1 and (5.86)-(5.104) in Subsection 5.4.2 are valid for pooled quantities. However, the
evaluation of second moments of X and y, based on probability models (5.4), (5.5) and
(5.42) may or may not be valid.
A
For example, we shall show that when the tagging strata are pooled, the variances of
are estimated correctly. By contrast, when the recovery strata are pooled, the variances are
overestimated unless the sampling probabilities are equal. To see this, first suppose that
two of the tagging strata are pooled under conditions stated in Result 6.1. Without loss of
generality, change the labels of tagging strata so that the pooled strata will be the last two.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 125
Now, to check the validity of variance formulae (5.83) for the pooled data, it is enough to
examine the validity of equations (5.79) and (5.81) for the pooled data.
It is easy to see that equation (5.81) is valid since only the tagging strata are pooled.
Following (5.79), the investigator is obliged to use
in place of Cjk p j p k E [ X v s ] , the (i, i)th element of E[X8pp'X*'] . Clearly, this is correct
for i = 1, . . . , s - 2, since those strata are not pooled. Let us check its validity for i = s - 1.
Note that for j = 1,. . . , t ,
Next note that for j # k,
Therefore, the (s - 1, s - 1)th element of E[X8pp'X*'] is
= x (m8-l0(,-l)j + mso(3)j) pj - (m3-l f ms). i
But, according to (6.8),
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 126
Hence, ~ j k p j p k E I X ~ , - l ) j X ~ ~ - l ) k ] is in fact equal to (6.34). This implies that when only
tagging strata are pooled under sufficient conditions, then, the variance formulae given in
Section 5.4.1 are valid for pooled data.
NOW, suppose that two of the recovery strata are pooled under sufficient conditions given
by Result 6.2. Without loss of generality, change the labels of recovery strata so that the
pooled strata will be the last two. Now, to estimate ~ a r ( ? * ) , the investigator is obliged to
use
in place of Eigi pjpkE[XGX:k], the (i , i)th element of E[X*$$'X*']. To check the
validity of this, note that
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
t - 2
-2mi x ~ j P t - 1 ( e i jP je i ( t -1 )~ t -1 e i j ~ j e i t ~ t ) - j=1
Now, recall from (6 .15) and ( 6 . 2 3 ) that under sufficient conditions,
Hence, the above can be written as
This implies that (5 .79) is correctly estimated if the pooling is done under the sufficient con-
ditions. However, we can show that (5 .81) is overestimated unless the sampling probabilities
are equal. To see this, note that the investigator is obliged to use
in place of E [ ( Y ; ) ~ ] . This is correct for j = 1 , . . . , t - 2 , since those recovery strata are not
pooled. To check its validity for t - 1 , note that
E [ ( ~ ; - l ) ~ ] = E [(Yt-1 + Yt)2] -
= V t - l P t - l ( 1 - pt-1) + V t p t ( 1 - pt ) . -
Recalling from (6 .19) that under sufficient conditions, @t-l = V t ~ t - l f " t p t one can show Vt-1 +Vt
that
This shows that the term E [ ( y t - , ) 2 ] is overestimated unless pt-1 = pt. So, the overall effect
of pooling recovery strata is to overestimate the variances unless the sampling probabilities
in the pooled recovery strata are equal. This result is consistent with Darroch7s finding that
the variance of the Petersen estimate will be overestimated unless the sampling probabilities
are equal.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
6.5 Biases of Pooled Estimates
In general, pooling can produce biased estimators. If the pooling is done under sufficient
conditions stated above, then the estimators are consistent and asymptotically unbiased.
To calculate the bias that can occur otherwise, first suppose that s* 2 t * , and 3 and n*
are of full rank. Then,
2 = (n8)tm* = (n8'n*)-'n*'] m*, and h -* A
I V = DV8p = ~ ~ * ( n * ) ~ m * . (6.36)
By repeating arguments in Subsection 5.4.1 with pooled quantities, we can show that
E (?) = Dg* (3'3) -' (h ) 'm8 + O(1)
= ~ g * ( i T ) + m * + O(1). (6.37)
To see this, first let
Then,
n* = 3 + X*. (6.38)
n8'n* = I" + Z*,
where
Now, using the identity (5.62), we can get
(n8'n*)-' = [I - r*-'z*]r*-' + ( r*- '~*)~(n* 'n*)- ' , and
(n*lt = (,*In*)-In*' = [I - I'*-'Z*]I'*-'(S' + x*') + ( I ' * - ' z* )~ (~*)~ . (6.41)
Now, Post-multiplying (6.41) by m*, substituting S I X * + X * ' 3 + X8'X* for Z* in the
square bracket, and arguing as in Subsection 5.4.1, it can be shown that
A
p = r*- ' i t 'm* + a random quantity with expected value ~ ( r n ; ' ) .
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
In other words,
Next, let
Then, from (6.36)
Since y and 5 are independent, and E(y8) = 0, it follows that
If the sufficient conditions stated in Result 6.3 are satisfied, then DV*(n*)trn8 = 7, h
and hence is asymptotically unbiased. Otherwise, it can be biased.
Now recall from (5.50) that V = D4ii) t rn. Therefore, actual expected numbers of
untagged animals in pooled recovery strata can be obtained by adding the corresponding - - * V in . Let this be denoted by (v) . Then, the asymptotic bias of Y is given by
DV*(iT)trn8 - (v)*.
The bias can be positive or negative depending on the movement patterns and sampling
probabilities. To see this consider the following hypothetical examples. Let
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
First, let
Then,
Pooling the last two tagging strata, and the last two recovery strata, we get
2000
* [ 20.0) 3 = [ I: i' ) , and V = ( 4400 ) . m = 11550
9500 78 157.5
Hence,
So, the bias of the total estimate is
Next, suppose that the sampling proportions in the last two recovery strata were inter-
changed. I.e,
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Then,
Pooling the last two tagging strata, and the last two recovery strata, we get
m * = "= [ Y I': 1, and V = ( 4 4 0 0 ) . 9950
9500 78 122.5
Hence,
So, the bias of the total estimate is
~ i a s ( c ) = (271077 + 381681) - (220000 + 430000) = 2758
Similar results can be shown for the bias of 6, when s* 5 t*.
6.6 Worked Examples and Monte Carlo Studies
6.6.1 Example 1: Schaefer's sockeye salmon data
As the first example, we illustrate our methods using a data set provided by Schaefer(l951).
Darroch also used this data set to demonstrate his methods. So, to compare results from
both approaches and to point out the advantages of our method, we reproduce main points
of Darroch's calculation as well.
In this experiment both stratifications are with respect to time instead of place. The
population consisted of adult sockeye salmon passing a certain point of a river in British
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 132
Columbia. The experiment was conducted by Schaefer. A total of 2,351 fish were tagged
on the way to their spawning grounds, over an 8-week period. Later, samples were drawn
regularly over a 9-week period as the fish spawned and died on the spawning grounds further
upstream: 10,472 fish, of which 520 had been tagged were recovered in these samples. The
data from the experiment are set out in Table 6.1.
Week of
tagging (i) mi
Total 2351
v.i
Week of recovery ( j )
1 2 3 4 5 6 7 8 9 Tot a1
3
11
76
180
183
60
6
1
Table 6.1: Schaefer's data on sockeye salmon: n;j, m; and vj for s = 8, t = 9 (from Darroch
[1961:Table 11)
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Darroch's approach
1. He reduced s and t to four by grouping the first three and last three weeks of tagging
into two single strata and the first three and last four recovery strata into two single
strata, since the n;j in some of the outer weeks were too small. The new values of
nfj, mf and v; are given in Table 6.2.
Week of
tagging (i) mf
Week of recovery (j)
1-3 4 5 6-9 Total ni;/mf
Table 6.2: Schaefer's data on sockeye salmon: nfj , mf and vj* for s* = 4, t* = 4 (from
Darroch [1961:Table 21)
2. He assumed that the 4;" are equal, and estimated p* through (5.6) as
He pointed out that the unsatisfactory value of may be just a symptom of the
general inadequacy of the estimators, or it may indicate that the model is incorrect in
assuming the 4; equal.
3. He argued that n>/mf indicate where the possible differences in the 4; lie. Since the
middle two are appreciably larger than the outer two (cf. Table 6.2), he suggested
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 134
imposing constraints, = 44 and d2 = 43, and using (5.12), and (5.12) to obtain.
estimates.
4. To evaluate (5.12), ADZ n should be a square matrix. To accomplish this, two more
recovery strata should be pooled. It is permissible to group two recovery strata if
R.6.2.1 (b) is satisfied. In Table 6.2, the only two columns that might satisfy this are
the third and fourth columns. He pooled those columns.
5. Equations (5.12) and (5.52) led to the following estimates.
Recovery
Stratum
Tot a1
Table 6.3: Darroch's Estimates of V*
Our approach
Now, we present our approach on Schaefer's data. We think that it is more meaningful to
estimate the numbers at the tagging stage rather than the numbers at the recovery stage.
The reason is that the survival probabilities are not equal to one. So, the estimated numbers
at the recovery stage would actually estimate the numbers that would have reached recovery
strata if the survival probabilities were one. However, in order to compare with Darroch's
approach, we also estimate the numbers at the recovery stage. The steps followed were:
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 135
1. Pool only the recovery strata. As Darroch did, we pool the first three and last four
recovery strata. Then, s = 8, t* = 4.
2. Derive the least-squares estimate (LSE) and an iteratively re-weighted least-squares
estimate (IRLSE) of p using (5.35) and (5.41) respectively. The least-squares estimate
is taken as the initial guess for deriving the weighted least-squares estimate. The mean
squared errors are estimated using (5.77). These estimates are shown in Table 6.4
Table 6.4: Our Estimates of V*
Recovery
Stratum
1-3
4
5
6-9
Total
Even though the least-squares method produced an unacceptable estimate of the sam-
pling probability in recovery stratum 4, the iteratively re-weighted least-squares estimate
is acceptable and may be reasonable too. High root mean squared errors indicate that the
estimates of individual stratum sizes are not reliable. However, the root mean squared error
of the total estimate implies that this estimate can be reliable.
Next, we present our estimation of the numbers at the tagging stage. For this, we pooled
only the tagging strata. The steps followed were:
1. Pool the same tagging strata as Darroch did; i.e, first three and last three tagging
strata. Then, s* = 4 and t = 9.
LSE IRLSE
P
0.1314
1.7899
0.2084
0.1005
A
P -
0.1562
0.4344
0.3054
0.0988
h
V*
5,422
6,133
10,860
31,625
54,040
F
6,444
1,488
15,914
31,055
54,900
&z@j 1,648
9,128
14,134
12,520
6,528
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 136
2. Derive the least-squares estimate (LSE) and an iteratively re-weighted least-squares
estimate (IRLSE) of U* using (5.35) and (5.43), respectively. The least-squares es-
timate is taken as the initial guess for deriving the weighted least-squares estimate.
The mean squared errors are estimated using (5.91). These estimates are shown in
Table 6.5.
Tagging
Stratum
Total
LSE 11 IRLSE
Table 6.5: Our Estimates of U*
Again, the high root mean squared errors indicate that the estimates of individual stratum
sizes are not reliable. However, the estimates of total seem to be more reliable. This
phenomena is common when the samples are not large enough.
6.6.2 Monte Carlo Studies
As an investigation of the accuracy of the above estimation for the total, we performed
three Monte Carlo studies. Since the estimated U+ in this problem is close to 50,000 fish
and the total number of tagged animals is 2351, the proportion of untagged fish to tagged
fish is close to 20:l. So, in our first simulation we matched this proportion with a little
variation. In the other two simulations we gradually decreased this proportion allowing
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
a large variation. But, the released numbers of tagged animals were doubled to produce
reasonably large numbers of recovered tagged animals. The steps taken in the simulations
are as follows:
1. Let m be the numbers of tagged fish released in Schaefer's data.
2. 4 = D d n was calculated from Schaefer's data.
3. Matrix 4 was augmented by including the ( t + 1)th column of probabilities ( 1 - 41) .
Let the augmented matrix be denoted by 4.
4. Let Y(a, b) denote a rounded uniform random number in the interval (a, b). In the
three Monte Carlo studies, the numbers of untagged animals at the tagging stage (U)
were simulated as,
U' = 20 m' + [Y(O, 50), Y(0, 50), Y(0, loo), Y (0, 2, OOO),
Y(0, 2,000), Y(0, 5, OOO), Y(0, 1,000), Y(0, 500), Y(0, 400)],
(b) U; = 2 mi x Y(lOO,3OO), and
(c) U; = 2 m; x Y(250,400)
5.. The recovered numbers of tagged and untagged numbers ( n and v) were simulated
according to a multinomial model with probabilities given by 4 and numbers at the
tagging stage as follows.
(a) the numbers of tagged animals = m , and the numbers of untagged animals = U
as in (a) above.
(b) the numbers of tagged animals = 2m, and the numbers of untagged animals =
U as in (b) above.
(c) the numbers of tagged animals = 2m, and the numbers of untagged animals =
U as in (c) above.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 138
6. The least squares estimates of the total numbers of untagged fish at the tagging
stage (U+) were calculated based on the generated data, using the same steps as in
estimation of Example 1 ( pooling first three and last three tagging strata, and so on).
X 100. 7. Percent relative error of the total was calculated as U+
8. Repeated steps 4-7 1000 times.
The histograms and box-plots of the observed percent relative errors of overall totals are
shown in Figure 6.1. The figures corresponding to three studies are labeled as (a), (b) and
(c) respectively.
(a)
Figure 6.1: Percent relative error of U+ in three Monte Carlo studies (each study is based
on 1000 simulations).
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 139
In the first study, the tagging proportions were close to 1/20. But the released numbers-
of tagged animals were rather small (as in Schaefer's data). Because of this, the nij's were
of the same sizes as in Schaefer's data set. In a number of simulations, the estimates of
individual stratum sizes were negative. However, on average, the total estimates seem to
be very reliable. It can be seen from the histogram (a) and the box-plot (a), that the
percent absolute relative errors of the totals are reasonably low. In the second Monte Carlo
study, the numbers of marked animals were twice the amount in Schaefer's data set. The
tagging proportions were smaller; between 1/100 and 11300. In the third study, the tagging
proportions were further reduced to proportions between 1/250 and 11400. In both studies,
a number of negative estimates of stratum sizes were observed. However, the total number
of fish at the tagging stage appears to have been estimated accurately with high probability.
6.6.3 Example 2: Schwarz's pink salmon data
Now we derive estimates using a data set provided by Dr. Carl Schwarz of the Department
of Mathematics and Statistics, Simon Fraser University. These data were collected in the
Fraser River, British Columbia. In this experiment, tagged male pink salmon were released
over 6 weeks. As the fish reached each of 6 spawning areas, they were again sampled over
several weeks. This produced 31 recovery strata, considering each recovery week at each
spawning ground as a stratum. The numbers of marked and unmarked animals released
and recovered are presented in Tables 6.6 and 6.7.
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 140
Area of
recovery
Table 6.6: Fraser River pink salmon data: Areas 1-3
Total number of
unmarked fish
recovered
9,004
9,279
5,984
3,287
745
1,299
8,374
18,843
19,401
8,975
916
5,837
3,960
1,710
29
Week of
recovery
2
3
4
5
6
3
4
5
6
7
3
4
5
6
7
Week of tagging
1 2 3 4 5 6
Number of marked fish released
3479 5354 4219 4227 1588 327
Number of marked fish recovered
14 15 7 3 0 0
5 12 7 3 0 0
0 4 4 6 1 0
0 5 1 4 0 0
0 0 2 0 1 1
0 1 1 6 1 1
0 2 6 22 15 1
0 1 7 29 38 2
1 2 5 23 42 10
0 1 1 8 17 11
0 3 3 0 0 0
0 5 7 7 7 1
0 0 1 1 5 2
0 0 1 3 2 0
0 0 0 0 0 0
Continued on next page
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
Total number of
unmarked fish
recovered
Table 6.7: Fraser River pink salmon data: Areas 4-6
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 142
From these data, it is more meaningful to estimate the numbers of fish at the tagging
strata than to estimate the numbers of fish at the recovery strata. There are two reasons
for this: The first is that Assumption 6 may be more reasonable than Assumption 5. The
second is that the number of tagging strata is smaller than the number of recovery strata
(s < t ) . Following is a list of the more noteworthy features.
1. In this data set, there are several recovery strata in which the numbers of tagged fish
are zero or very small. It may be reasonable to assume that within each recovery area,
the sampling proportions are kept roughly equal. Then, it is permissible t o pool strata
within areas. So, we pool recovery strata as follows:
Area 1: weeks 4-6, Area 2: weeks 3-4,
Area 3: weeks 3-4, and 5-7, Area 4: weeks 4-5,
Area 5: weeks 1-2, and 5-6, Area 6: weeks 1-2, and 4-7.
The pooled data set is shown in Table 6.8.
Week of
tagging (i)
Area of Recovery
Table 6.8: Fraser River pink salmon data: Recovered numbers of marked fish ( after pooling).
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census 143
2. The least squares estimates obtained using the pooled data is shown in Table 6.9. One
of the estimates is unacceptable since it is negative. Such an unacceptable estimate
Table 6.9: Estimated numbers of unmarked fish at the tagging stage
Week of tagging (i)
1 2 3 4 5 6
can be a result of incorrect modeling or near singularities of the matrix n. Table
6.8 suggests that there may be a collinearity between the second and third rows of
corresponding 6 matrix (p-value =.12). So we pool those two tagging strata. In
addition, the counts in the last row are very small except in Area 2. So, we pool the
last two tagging strata as well. The least-squares and weighted least-squares estimates
Tot a1
obtained using pooled data are presented in Table 6.10
Week of
tagging
Total
LSE IRLSE
Table 6.10: Estimated numbers of unmarked fish at the tagging stage
Chapter 6. Pooling in a Stratified Two-Sample Tag Recovery Census
6.7 Discussion
In stratified tag recovery experiments it is common to pool of some or all of the strata. The
estimate of the population size in such a case may be inconsistent. Darroch [13] has described
conditions for complete pooling which are sufficient for the resulting pooled estimate to be
consistent.
We discussed the case of partial pooling and suggested conditions which are sufficient
to preserve the consistency of the estimates. Conditions 1 and 3, proportional tagging
and proportional sampling are potentially a t the investigator's control, although they are
difficult to implement without prior knowledge of the population sizes. However, preliminary
estimates of proportional sizes of strata may be obtained from other sampling methods such
as the 'CPUE' (catch per unit effort) method. Then it may be possible to maintain the
conditions of proportional tagging and proportional sampling.
Naturally, the Conditions 2 and 4, which are on the movement probabilities, are not
at the investigator's control. However, when they are satisfied, it is possible to detect the
presence of collinearity caused by these conditions. Then the corresponding strata may be
pooled to obtain consistent estimates.
If these conditions are met, the investigators can use the pooled data to derive estimates
of the stratum sizes. However, the variance estimators provided for unpooled estimates are
not necessarily be valid for the pooled data. For example, as Darroch showed that when
all the tagging and recovery strata are pooled, the variance can be overestimated unless the
sampling probabilities (the pj's) are equal. We showed that under partial pooling of tagging
strata, the variance of ?* is correctly estimated using formula (5.83) with pooled data. We
also showed that effect of pooling recovery strata on this formula is to overestimate the
variances unless the sampling probabilities are equal in the pooled strata. We were unable
to determine the effect of pooling on the variance formula for 6, because of its complicated
form. This problem is open for further research.
Appendix: Some Useful Results
In this appendix, we present some o f the important results that were used throughout the
thesis.
According t o Sen and Singer [34](pg. 43) ,
Theorem 1 (Jensen's inequality) Let X be a random variable, and g ( x ) , x E R, be a
convex function such that E [ g ( X ) ] exists. Then,
with the equality sign holding only when g is linear almost everywhere.
According t o Marsden and Tromba [26](pg. 274),
Theorem 2 Let f : U C R~ -+ R and g : U C R2 -+ R be smooth functions. Let
vo E U, g (vo) = c, and S be the level curve for g with value c. Assume that V g ( v o ) # 0
and that there is a real number X such that V f ( v o ) = XVg(vo) . Form the auxiliary function
h = f - Xg and the bordered Hessian determinant
evaluated at vo.
1. If > 0, then vo is a local maximum point o f f restricted to S .
145
Appendix: Some Useful Results
2. If < 0 , then vo is a local minimum point o f f restricted to S .
3. If = 0 , the test is inconclusive and vo may be a minimum, a maximum, or neither.
According to Kotz and Johnson [21](pp. 386-387 ),
Theorem 3 (Cauchy-Schwarz inequality) Let ( X , Y) be a bivariate random variable.
Then,
E ( x ~ Y ~ ) E ( x ~ ) E ( Y ~ ) ,
provided the expectation on the left hand side exists.
According to Kotz and Johnson [22](pg. 515),
Theorem 4 (Slutsky-Frechet) If the sequence of random variables {X,) converges in
probability to a mndom variable X , then so does f (X,) to f ( X ) for any continuous function
f .
According to Albert [l](pg. 30),
Theorem 5 xo minimizes
if and only i f xo is of the form
xo = H ~ Z + ( I - H ~ H ) ~
for some y. The value of x which minimizes llz - Hx1I2 is unique if and only i f H t H = I .
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